Mr Tom Evans and Ms Kate James had made an application to the Supreme Court to have the case re-examined after exhausting all other legal appeals for their 23-month-old son Alfie Evans, who is being treated at a hospital in Liverpool, northwest England.
In appealing the lower court rulings, Alfie's parents, Tom Evans, 21, and Kate James, 20, argued their son had shown improvement in recent weeks.
The hospital will begin end of life care for Alfie Evans. The hospital must be free to do what has been determined to be in Alfie's best interests. Crowds of supporters also besieged Alder Hey Hospital to demonstrate and protest on Alfie's behalf.
'As we explained in our earlier decision in this case, the best interests of the child are the "gold standard" which is not only adopted by our law but also reflects the global standards to which this country is committed.
Earlier court rulings blocked further medical treatment and ordered life support to be withdrawn. But that it has happened and is continuing to happen can not be denied.
"It means that Alfie can not breathe, or eat, or drink without sophisticated medical treatment".
It means "there is no hope of his ever getting better", the judges added.
The father of a terminally ill British toddler met Wednesday with Pope Francis and begged him to make another bid for his son's life, after an appeals court ruled that the child can not be taken to Italy for further medical treatment.
Judges said no detail of the plan could be made public because Alfie was entitled to privacy as his life came to a close.
On Saturday, Thomas Evans released a statement saying he has filed an appeal - the family's second - with the European Court of Human Rights.
Baroness Hale of Richmond said that it was the view of doctors at Alder Hey Children's Hospital, Liverpool, that nearly all of the 23-month-old boy's brain had been destroyed. Doctors at Alder Hey have said life-support treatment should stop because further treatment was futile.
The couple said Italian doctors were willing to treat the little boy and an air ambulance was available. He said flying Alfie to a foreign hospital would be wrong and pointless.
Appeal judges upheld Mr Justice Hayden's decision.
"We have instructed our lawyers to submit an urgent application to the European Court of Human Rights, and they have done so today".
The Supreme Court had also dismissed the suggestion that Alfie was either "detained" or "unlawfully detained". It is a piece of common law which probably dates to Anglo-Saxon times.
Mr Justice Hayden dismissed that habeas corpus claim last week.