The couple were refused permission to appeal a High Court ruling
A seriously ill baby, whose parents lost a court battle to keep him alive on a ventilator, has died.
The nine-month-old, known as "Baby OT", had a rare metabolic disorder and had brain damage and respiratory failure.
His parents had appealed against a ruling at London's High Court that it was in the boy's best interests to withdraw "life-sustaining treatment".
The British Medical Association (BMA) supported the doctors' actions, as did some Christian groups.
Baby OT was unable to breathe by himself and died at 1008 GMT after doctors withdrew his treatment.
His parents said they were "deeply distressed" by the decision and said the life of their "beautiful boy" was worth preserving.
'He died peacefully'
The couple said through their solicitor after the death was announced: "During his short time with us, OT became the focus of our lives. We were present during his last moments, together with [his] extended family.
"He died peacefully. We will miss him greatly and wish to say that we are proud to have known our beautiful son for his brief life."
Doctors treating him had said the boy's life was intolerable and his disability was such that his life had little purpose.
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However his parents argued before he died "he experiences pleasure and that he has long periods where he was relaxed and pain free".
The High Court ruling on Thursday gave doctors at an unnamed NHS trust powers to turn off the ventilator that kept Baby OT alive.
On Friday two Court of Appeal judges refused the couple permission to challenge the decision by Mrs Justice Parker made after a 10-day hearing.
The baby, his parents and the hospital trust involved in the case cannot be identified, as part of the court ruling.
Lord Justice Ward was told the couple had decided to wait outside the courtroom while the ruling was given as they could not face hearing the decision.
A spokeswoman for the BMA said: "Cases like this are very distressing and we have every empathy with the parents, but when the parents and the clinical team don't agree on the treatment for the child in question, the only way forward is to go to the courts and for the courts to decide on what is in the best interests of the child, which is paramount."
The Roman Catholic Archbishop of Cardiff Peter Smith told the BBC: "We must not kill an innocent person under any circumstances, there is no moral justification for that.
"But when it comes to the end of life, we wouldn't take a vitalist view, and say somebody has got to be kept alive by most extraordinary means."
My heart goes out to the family, but also to the medical team
Dr Peter Saunders, general secretary of Christian Medical Fellowship, which represents some Christian doctors in the UK, said: "This tragic case highlights both how precious human life is but also that there are limits to medicine.
"Withdrawal of treatment decisions can be agonisingly difficult requiring great wisdom and sadly differences between parties can sometimes only be resolved by the courts.
Professor Sam Leinster of the University of East Anglia school of medicine, told the BBC: "My heart goes out to the family, but also to the medical team.
"The court is charged with weighing all the evidence that is put before them.
"There is a belief among us, it's kind of fed by everyone around us, that medicine has the answer to everything. But unfortunately there are occasions where medicine can't do anything."
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