When Esther and Dan Levy’s son Andrew was 14 months old, he received a diagnosis of a kind of leukemia so rare that their medical team said getting it was like being bitten by a shark and struck by lightning at the same time.

Leukemia, a cancer of those cells in the bone marrow that produce new blood cells, has many varieties, but the most common type in children, acute lymphocytic leukemia, is largely curable. Andrew’s cancer, however, a subtype of acute megakaryoblastic leukemia (AMKL), affects only about 45 children a year nationwide and is much more difficult to treat. The odds of surviving this type of AMKL are roughly even — unless the child is one of a handful who happen to have a particular genotype, in which case these odds plummet to a mere one in 10. Genetic analysis revealed that Andrew was in this tiny group.

There was more bad news. Two weeks after the diagnosis, Andrew’s doctor, Norman Lacayo, an oncologist at Lucile Packard Children’s Hospital at Stanford University, received an urgent call from Michael Loken, the president of Hematologics Inc., a Seattle lab that was analyzing Andrew’s cells. Loken had recently discovered that a small percentage of children with AMKL had a specific phenotype — a pattern of proteins on the surface of the leukemia cell he called R.A.M. (a former patient’s initials) — that independently predicted a terrible outcome, with a survival rate of about one in six. Andrew had this phenotype too.

“Has anyone ever survived this kind of cancer?” Dan asked Lacayo. “All I wanted to know is that it was not impossible,” Dan recalls. Lacayo said yes, but Dan felt his answer was “foggy.” The truth was that the team couldn’t find a single equivalent case in the literature.

Beginning on that December morning in 2014 when Esther took Andrew to the E.R., she recalls, she felt as if they had stepped into a horror movie, the unfolding events both surreal and evil. Up to that point, Esther and Dan had led, in her words, “charmed lives — picture perfect.” Only a small subset of people would sincerely say that nothing truly bad has ever happened to them; before the diagnosis, Esther and Dan say, they were among them. When Andrew got sick, they were in their mid-30s and energetic, optimistic and extroverted. They had both attended Stanford — Dan majored in industrial engineering, Esther in human biology, with a minor in dance — before going on to successful careers. Dan founded a sports-related start-up, then became vice president of small business at Facebook, while Esther worked at Kurbo, a start-up focused on weight management for kids, and taught spin classes at a Jewish community center for fun. Their own families were stable and close-knit; to recall any true adversity in either family, they had to think back to a grandmother of Dan’s whose family perished in the Holocaust.

Once Andrew’s illness was diagnosed, he needed a bone-marrow transplant as swiftly as possible. First the doctors had to kill the leukemic cells in Andrew’s bone marrow with chemotherapy, then replace them with a donor’s cells. Andrew’s 3-year-old sister, Lea, and his 5-year-old brother, Wills, were tested, and in the family’s first bit of luck since the diagnosis, Wills turned out to be a perfect donor match. Andrew underwent two rounds of chemotherapy, but there were still traces of cancer when the transplant was performed in February 2015, putting the outcome at high risk of failure.

The Levys had created a Helping Hands website, where friends signed up to host play dates or deliver meals (as did our family because our children were in the same school as Wills), and a Facebook group for updates on Andrew’s illness, which 1,700 people joined. But despite all the support, Esther felt deeply “alone with the experience,” she says. Her former life had vanished: She was living in Andrew’s hospital room, sleeping on a sofa that opened into a hard bed. She had left her job and the rest of her family while Dan continued to work and live at home with Wills and Lea. Her nights were punctured by Andrew’s cries; her days were spent frantically trying to distract him from his pain and nausea, cleaning up his vomit, holding him down during blood draws and making stressful medical decisions. He screamed if she left him for a few minutes, even to use the bathroom or shower.

After Esther and Andrew spent three months in the hospital, the entire family moved into a nearby apartment, in order to live in a smaller space they could keep immaculately clean while waiting for Andrew’s new immune system to develop. Esther remained Andrew’s full-time nurse, responsible for a dizzyingly complex regimen of medications and sterile changes of the IV. A bone-marrow test that April showed no traces of cancer, and Andrew was considered to be in remission. They posted videos of him banging on his drums and singing with his toy Elmo and pretending to play golf.

They decided that when Andrew was well enough, they would not return to their old home but begin a new life. They found a house in the nearby town of Atherton in the style of an English country manor, encircled by hedges and white rose bushes, that suggested privacy and safety. Andrew was too vulnerable to leave the apartment, so Esther could not go to see the house in person, but they bought it anyway, and she made plans with a decorator friend to create an airplane-themed room for Andrew.

But on June 19, the medical team told Esther and Dan that there was bad news again: Andrew’s cancer had returned. The number of cells was small but would inevitably grow, the doctors explained. The team presented a new plan: They would begin chemotherapy again in preparation for a second bone-marrow transplant, perhaps using cord blood this time.

“Oh, God,” Esther said, putting her head in her hands. She felt she could not go through it all again. And there was no reason to think it would work. The odds of success during the first transplant had been long; in a second attempt, they would be much more so. “But the odds that it would cause all of us more suffering were 100 percent,” she told me.

From the initial diagnosis, Dan had determined that their goal was not simply to help Andrew survive but to keep the family intact. To choose to move back into the hospital, where they believed Andrew would die, was “a fundamental violation of every promise we made to ourselves and our kids that we would be together again,” he told me. He felt the family had just started to heal from the months of separation. “The emotional scars of the experience,” he said, “would be irreparable if we ripped them open and split our family apart again.” They decided to stop treatment. They would move to their new house, where Andrew would spend whatever time he had splashing in their swimming pool and playing in the grass with Wills and Lea.

The doctors were stunned. “We love you, and we love Andrew and we’re not ready to give up,” Jennifer Willert, the pediatric oncologist in charge of the transplant, blurted out. Lacayo and Willert argued for at least trying some palliative chemotherapy to prolong Andrew’s life. Esther and Dan hesitated but ultimately declined. They called their decorator friend and told her to return the furniture for Andrew’s new room. She was one of the first people to whom they told the news: Andrew was going to die.

The Levys posted the news of their decision on June 22. They explained their thinking and asked their friends not to question their choices, recommend new treatment options, tell them about God’s plan or insist that there was hope. “I truly believe that I have a new way of looking at parenting — it is not about the length of life that matters, but the quality of life,” Esther wrote. “We are going to focus on quality.”

But quality time with a doomed child turned out to be impossible. The cancer cells were few enough that they were not yet making Andrew sick, but, Esther posted, “I can’t think of anything more painful than spending time with your precious baby knowing that he is going to die soon.” Parenting is teleological; parents rear a child to become an adult. What were their goals for Andrew now? “I am no longer ‘raising’ him to grow up to be a wonderful human being,” Esther wrote. Should she let him eat junk food or watch videos on the iPad all day? Did it matter?

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Their older kids asked tormenting questions. Lea wanted to know whether they could buy Andrew a certain toy when he was 4 years old like her. Wills wanted to know why they had Andrew if they knew he was going to be sick all the time.

Dan read them Mo Willems’s book “Waiting Is Not Easy!” about an impatient elephant. As he read, he thought about the waiting that had engulfed them over the previous nine months. They had waited to get the right diagnosis; they had waited 100 days for the transplanted cells to grow; they had waited for the results from the bone-marrow tests to see if the cancer was gone. “Now there are no more diagnoses,” Dan wrote on Facebook. “No more tests. And no more milestones. But there is waiting. Maybe hours, or days, or weeks.” This was the most agonizing of all: the wait for Andrew’s death.

On July 1, they moved into their new house, and Andrew became sick. By the holiday weekend, he was moaning or screaming in pain whenever he was awake. Dan took a leave of absence from work. Esther held Andrew at all times, his body draped over hers on the couch or the bed. Dan took food to her because she couldn’t hold him and sit up at the dinner table. Her hair began to fall out because of the stress. “It was unbearable for him and for us,” she says.

The hospice team began to come every day to try to control the pain with high doses of opioids. Harvey Cohen, an oncologist and the medical director of the hospital’s palliative-care program, explained to them that as the disease progressed, Andrew would not have enough platelets for his blood to clot. A hospice nurse told them to buy dark towels for Andrew’s crib, so that if he started to bleed uncontrollably, the sight would be less frightening for his siblings and for them.

During the second week of July, the hospice team told them to prepare for Andrew’s imminent death. They called a rabbi, and thinking about how Andrew loved airplanes, they picked a Jewish cemetery near the airport. Not wanting him to be buried alone, they purchased grave sites for themselves as well. They established an Andrew Levy Memorial Fund to raise money for music therapy at the Lucile Packard Children’s Hospital.

The members of their medical team visited their home to say goodbye. Andrew had stopped eating. He was barely moving, his breathing raspy and his complexion sallow, with the particular look the team knew from other dying children. Sometimes he stopped breathing momentarily, and his body would become rigid, and his face turn blue. “It’s O.K. for you to go,” Esther told him. All she wanted now was for this to end quickly.

They called Wills and Lea into the living room — a room the kids rarely entered. Esther pulled them close to her on the couch, and Dan sat on a cushion on the floor. They had rehearsed what they were going to say with Barbara Sourkes, a hospital psychiatrist with whom they had grown close, and they made an audio recording of this moment in case they needed to discuss it with her later.

Dan told the children that the transplant had been a success, and that Wills’s cells had done a great job, but that Andrew’s cells needed to work on their own at some point, and they weren’t. “His body is just not working,” he said, as straightforwardly as he could manage.

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“Is Andrew going to get better?” Wills asked.

“The doctors don’t think so, Wills. No.”

Sourkes had advised them to tell the children only what they needed to know so as not to overwhelm them, because the children needed emotional space to process things their own way. “So Andrew — Andrew is going to die at some point,” Dan said. “We don’t know when.”

“I don’t like that Andrew is going to die!” Lea exclaimed and started crying.

Wills pulled the hood of his sweatshirt over his face and said he didn’t want to talk about it.

“Andrew is going to die, so that means we are only going to have four people in our family,” Lea said unhappily. She asked if they could get a new baby to replace Andrew, and she and Wills began to fantasize about a new baby who would make everything all better.

Esther returned to Andrew. “I promise, I promise you, we are not going to forget him,” she said. “You are always going to have a brother named Andrew because he is always your brother, now and forever.”

“Andrew’s pieces of love will always be in our heart,” Lea said, and then they all agreed to watch Mickey Mouse together.

The vigil stretched on through the summer, and what they called “mirages” began to appear. In late July, Esther was sitting outside with Barbara Sourkes, holding Andrew and watching Wills shoot baskets. Suddenly Andrew sat up and reached for a ball and managed to throw it through his own little basketball hoop. Esther and Barbara were speechless.

At first the mirages were brief — Andrew would laugh when Lea showed him her bellybutton or would stack blocks for 10 minutes — and then he would lapse back into pained lethargy for the rest of the day. But soon these episodes began to lengthen. For Esther, the mirages did not feel like miracles but “evil tricks.” She went through intense surges of anger. “I felt like, How many trials are we going to have to endure?” she says. “Are we being spared nothing?”

Esther started sending the medical team videos. “Andrew is eating pizza, Andrew is sitting up, Andrew is laughing,” Lacayo, their oncologist, recounts. “And we are like, What?”

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In August, as the team struggled to account for what was happening, they theorized that in July, when everyone assumed Andrew was dying of cancer, he must have had a terrible infection instead, which passed. It didn’t change the prognosis: The doctors stressed that, while Andrew might continue to recover from that infection as his new immune system took hold, the cancer cells were also growing and would eventually overwhelm him.

After a blood test showed that his platelets were low, Cohen, the palliative-care doctor, urged them to accept transfusions to increase Andrew’s platelets so that, even though he was going to die, it would not be from bleeding to death. But at the hospital, it turned out, mysteriously, that Andrew had more platelets than at his last blood test, so there was no need for a transfusion that day. When Dan suggested giving him vitamins, Esther snapped at him. He seemed to be taking the anomalous blood test to mean Andrew was getting better, when, she says, “I had no hope, and I needed not to have hope in order to function.” And then they both apologized.

In September, Andrew began to walk again, and his appetite and energy and dark curls grew. Dan decided to return to work. Andrew turned 2 — a birthday his parents had never thought he would reach and knew would be his last. Esther recalls how friends urged them to enjoy every moment, and how she would tell them: “No, this is hell, and it sucks. He is still going to die, so there is nothing joyous about this time.”

When they first got Andrew’s diagnosis, she told a night nurse that she just wanted to get her happy-go-lucky little boy back for a single hour. She had not understood then that any reprieve would only mean that they would have to go through losing him all over again — “and each return will be harder than the last as Andrew grows and bonds with us,” she wrote in a post.

By October, Andrew was healthier than he had been in a year, running and playing ball with his siblings. None of the doctors had ever seen this kind of recovery before. They decided to bring him back to the hospital for a bone-marrow test.

Michael Loken, who had analyzed Andrew’s blood work, had not been surprised that Andrew’s cancer returned. He had been working on a paper about R.A.M., the genetic marker that Andrew had. He had tracked 19 other cases of children with the phenotype; three years after the diagnosis, only two were still alive and healthy. When he examined Andrew’s marrow this time, using a sample of 200,000 cells, he got goose bumps. He repeated the test with 500,000 cells. Then he called Lacayo with the news. The cancer had disappeared.

How could cancer spontaneously disappear? “It does feel a bit like a miracle,” says Jennifer Willert, the transplant doctor, echoing the sentiments of others. Noting the rare evocation of a concept that stands outside science, Loken says: “It certainly defied our expectations with no discernible basis of happening. I guess this may be the definition of a miracle.”

The medical team grasped for a scientific explanation. Because Andrew had received no treatment over the summer, the answer had to lie in the bone-marrow transplant of Wills’s cells. Their main theory was that the infection that nearly killed Andrew in July had triggered a huge increase in his new white blood cells — and that heightened immune response had attacked not only the infection but the cancer cells as well.

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The doctors theorized that the response was partly a product of timing: The cancer had returned just as Andrew’s new immune system grew strong enough to destroy the cancer cells. A critical part of why transplants work is that some of the white blood cells, the T cells, that grow from the transplanted bone marrow will attack any lingering cancer cells, an effect known as graft versus leukemia. Chemotherapy rarely kills every last cancer cell, so it is believed that without graft versus leukemia, the cancer will eventually grow back. This is often spoken of as a model of so-called immunotherapy — stimulating the patient’s own immune system to attack cancer cells — which is widely regarded as one of the most promising avenues for cancer treatment.

Willert had made a key decision to depart from Stanford’s protocol to increase Andrew’s chances of getting a robust graft versus leukemia effect. Typically, a leukemia patient receives immune-suppressing drugs for at least 100 days (and often much longer) in order to avoid a serious side effect called graft versus host disease, in which new T cells attack not only the cancer cells but also the patient’s skin, liver and gastrointestinal tract. The art of a transplant is said to be maximizing graft versus leukemia while minimizing graft versus host.

Willert, who is now at the University of California, San Francisco, Benioff Children’s Hospital, had advocated a rapid early taper of Andrew’s immune-suppressing drugs on Day 60, as is the practice at U.C.S.F. and other places, because she felt that the benefits outweighed the risk of graft versus host. “I fought for it because I have seen the power of getting rid of immune suppressants and letting the cells do their job,” she says. “After all, that’s the whole point of a transplant!”

The final, critical decision was made against medical advice: Esther and Dan’s resolution to stop treatment and let Andrew die. Had they permitted more chemotherapy, the treatment would have killed Wills’s cells, which were what ultimately enabled Andrew to live.

“When you have a child with a life-threatening illness, you have an irrevocably altered existence,” Barbara Sourkes had told the Levys, and Esther feels that is true. She had always felt in control of her fate, but now she believes this to be a fiction. She finds it difficult to reconcile bitterness over the blight of Andrew’s illness with gratitude for the reprieve. “We are the luckiest of the unluckiest people in the world,” she says. “I truly believe that.” The story presents itself to her as a riddle that cannot be resolved. She recalls her anger when others told them to hope. Is the lesson that their friends were right and there is always hope? Yet it was only by letting go of hope and accepting Andrew’s death that he lived.

She has not returned to work. “My full-time job is to help the kids feel safe again,” she says. But it is hard for her to feel safe. The two years after a transplant are the riskiest time for a relapse; after two years that likelihood plummets, and after five years, a patient is considered cured. The two-year mark is still nine months away.

“There are only two states after such a diagnosis: disease and uncertainty,” Cohen had told them. “Either he will die soon, and that’s certain — or he will continue on, and you will live with that constant balance of hope and fear. But the balance will change as time goes on.”

Only in the past few weeks, Esther says, has she been able to feel that she isn’t testing fate by scheduling a dentist appointment for Andrew six months out or by feeling moments of joy watching him without being shadowed by fear of the future. “Day by day,” she says, “we are allowing ourselves to celebrate a little more.”