The couple were refused permission to appeal a High Court ruling
The parents of a seriously ill baby have said they are "deeply distressed" by a court ruling allowing him to die.
They said they planned to "enjoy what little time" they had left with their "beautiful and beloved boy".
He has a rare metabolic disorder and has suffered brain damage and major respiratory failure.
The parents failed to overturn an earlier ruling and it is understood doctors will stop treating the nine-month-old within 24 hours.
"Belief in his humanity" was the reason they had fought the medical advice that he should be allowed to die, they said.
'Life is worthwhile'
Two appeal judges have upheld a High Court ruling which gives doctors at an unnamed NHS trust powers to turn off the ventilator keeping "baby OT" alive.
There is no further avenue of appeal for the parents.
The couple had sought to appeal against a judge's ruling on Thursday that it was in the boy's best interests to withdraw his "life sustaining treatment".
In a statement, they said they knew of only one other child with their son's condition and everyone was in "unknown territory".
"We are and always will be convinced that despite his desperate problems his life is worthwhile and is worth preserving as long as it is possible to do so without causing him undue pain.
"That was the real argument between us and the doctors - they think his life is intolerable and that his disability is such that his life has little purpose; but we, along with some of the nurses, believed that he experiences pleasure and that he has long periods where he was relaxed and pain free.
"Our belief in his humanity and inherent worth justified us taking every step to support him."
"Despite all the problems and creeping mistrust" between them and the medical team, the couple said they "remain enormously grateful to the National Health Service for the huge effort and massive cost that has been involved in OT's battle for life".
The Court of Appeal judges refused the couple permission to challenge a decision by Mrs Justice Parker made after a 10-day hearing.
Neither the baby, the trust involved in the case, nor the parents - Mr and Mrs T - can be identified.
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.
He said he would like to have addressed them personally and asked their lawyers to pass on the message that it was impossible not to feel the "deepest sympathy for their predicament".
The NHS trust had argued that the boy was suffering intolerable pain as a result of his treatment and condition and had no prospect of recovery.
Lord Justice Ward and Lord Justice Wilson said they would give the reasons for their decision at a later date.
Raanan Gillon, professor of medical ethics at Imperial College London, said everyone involved took the right course of action.
He told the BBC: "Rather than, as in the old days, the doctors would simply have said 'well, we know best and we withdraw the treatment', presumably they have tried hard to discuss and come to agreement about what's in the child's best interests.
"But given the lack of agreement they did the right thing, it's now the normal thing to do, which is to go to the courts - society's referee, as it were - to say look, we both want the best for this child, but we disagree radically about what's in the child's interest... it's your role to make a decision, and that's what's happened."
And Peter Smith, the Catholic Archbishop of Cardiff, believes the court's decision was a moral one.
He told the BBC: "There comes a point towards the end of our life when treatment becomes futile or incredibly burdensome, and causes suffering.
"At that point then, the church's view on that, the moral view is, that a patient or those acting on behalf of a patient can say, let nature take its course."
'Most awful feeling'
There have been relatively few similar cases in recent years.
In March 2005 a High Court judge ruled a 19-month-old boy, known as Baby MB, could be kept alive despite doctors telling a hearing his quality of life was so poor he should be allowed to die.
In that case the baby could not breathe unaided, swallow or chew but did not suffer from brain damage. The judge at the time said Baby MB was not unconscious and was capable of bonding with his family.
In February 2006 a judge ruled doctors would not have to resuscitate toddler Charlotte Wyatt, from Portsmouth, who was born prematurely with severe brain and lung damage.
In another case a six-year-old girl with an incurable brain disorder died before her case went to court. Amber Hartland from Cwmbran, Torfaen, suffered from Infantile Tay-Sachs, which left her almost completely paralysed.
Doctors had told her parents she would not be admitted to intensive care again and a judge would have to decide on her future care.
Ruth Winston-Jones went through a similar experience five years ago with her son, Luke, who was born with a genetic disorder called Edward's syndrome.
She too went through a legal battle but Luke died after High Court judges ruled that doctors could withhold mechanical ventilation.
"It's just the most awful feeling that anyone can imagine," said Ms Winston-Jones, from north Wales.
"The pain is the worst pain anyone can ever imagine. I worshipped and adored Luke and I've never had any more children since."