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Jim Wilson/The New York Times

Nailah and Martin Winkfield, Jahi McMath's parents, on Friday. A judge mediated an agreement between the family and Children's Hospital Oakland.

SAN FRANCISCO — It started out as a routine operation to treat an increasingly common medical problem in America, childhood sleep apnea. It became an anguished fight over the fate of a 13-year-old girl who, though pronounced legally dead by doctors, remains alive in the opinion of her religious parents.

The girl, Jahi McMath, was declared brain-dead after complications from surgery on Dec. 9 at Children's Hospital Oakland, which wanted to remove her from a ventilator. But her family protested in court, and she has remained connected to the machine.

On Friday, amid acrimonious battles in three courts, an Alameda County Superior Court judge mediated an agreement that could allow the child to be moved to another facility willing to take her, even though the hospital has declared her dead.

As arguments in the courts continue, the girl will remain connected to the ventilator at least until Tuesday, under the Superior Court judge's order. In the meantime, family members are scrambling to identify a facility that will accept the girl and doctors willing to carry out procedures that will keep her heart beating during the transfer.

Nailah Winkfield, the girl's mother, said she was hopeful that Friday's agreement would facilitate her daughter's move.

"I believe in God, and I believe that if he wanted her dead, he would have taken her already," Ms. Winkfield, a Baptist, said by phone. "Her heart is beating, her blood is flowing. She moves when I go near her and talk to her. That's not a dead person."

Jahi was admitted to Children's Hospital last month, and underwent three surgical procedures that included removing her tonsils and adenoids. She subsequently "suffered serious complications" that resulted in her death, according to court documents submitted by the hospital. The family's lawyer said in a court filing that Jahi suffered "large blood loss and, as a result, she suffered a heart attack and a loss of oxygen to her brain."

The hospital determined two days later that the girl was legally dead, and later sought to remove the ventilator that supplied oxygen to her body as her heart kept beating. The family objected, asserting that the heartbeat was proof that she remained alive. In a document filed in Federal Court, the family's lawyer stated that the girl's parents are "Christians with firm religious beliefs that as long as the heart is beating, Jahi is alive."

Doctors who have examined the girl, including a court-appointed neurologist, have all declared her brain-dead. A California Superior Court judge issued a temporary restraining order blocking the hospital from removing her from the ventilator. The order was extended until Jan. 7.

Judges in three separate courts have turned down the family's request to order Children's Hospital to insert tracheotomy and feeding tubes so she can be transported to another facility. Hospital officials have said that they would not allow the procedures to be carried out on the premises.

The family has attacked the hospital, saying it is effectively standing in the way of Jahi's move to another facility.

"We just want them to get out of the way," said Omari Sealey, the girl's uncle.

Christopher Dolan, the family lawyer, said a few facilities had agreed to accept Jahi, including the New Beginnings Community Center in Medford, N.Y. The center, which did not return calls seeking comment, said on its website that Jahi "has been defined as a deceased person yet she has all the functional attributes of a living person despite her brain injury." It did not say whether it was prepared to admit the girl.

Under the agreement reached Friday, the hospital will release the girl to the Alameda County coroner, a move that will officially classify the girl as dead, before the family can transfer Jahi elsewhere.

"It is hard for a mother to receive a death certificate for a child who has a heart beating," Mr. Dolan said. "It's an awkward situation."

The agreement does not require the hospital to perform the procedures that will help the girl's heart to keep beating during the transfer, or to allow an outside doctor to carry them out on hospital premises, as Mr. Dolan had requested.

"This is an important medical and legal victory for our hospital and all medical facilities, that they can't be forced to perform surgery," said Sam Singer, a spokesman for Children's Hospital.

Lt. Riddic Bowers of the Alameda County Coroner's Bureau said that once the bureau issued a death certificate and a release form, a deceased person's relatives were free to dispose of the body according to their wishes.

David Magnus, director of the Stanford Center for Biomedical Ethics, said it was unlikely that a long-term care facility would accept a patient who has been issued a death certificate.

"There's no reported case of a correct diagnosis of brain death where anybody comes back," he said. He added that while death occurs after an hour because of a lack of blood flowing into the brain, Jahi has not had blood flowing into her brain since at least Dec. 12.