Tom Evans and Kate James, who are both in their early twenties, had made another application to the court after losing a second fight over their son at the Court of Appeal.
"Alfie looks like a normal baby, but the unanimous opinion of the doctors who have examined him and the scans of his brain is that nearly all of his brain has been destroyed", the court said.
"No one knows why".
"It means that Alfie can not breathe, or eat, or drink without sophisticated medical treatment", the judges continued. It means that Alfie can not breathe, ore eat, or drink without sophisticated medical treatment.
It means "there is no hope of his ever getting better", the judges added. We affirm our conviction that all those who are and have been taking the agonising decisions regarding the care of Alfie Evans act with integrity and for Alfie's good as they see it.
Bishop John Sherrington said the bishops had a "deep awareness of the pain" for Alfie's parents. "He isn't diagnosed", adding that Alfie had been "left" on a feeding tube for 15 months.
Pope Francis met with Thomas Evans on Wednesday and conveyed his desire to the President of Bambino Gesu hospital through the Vatican's Secretary of State, Cardinal Pietro Parolin, that he wanted her to do everything possible to bring Alfie Evans to Rome for treatment. But doctors said his condition was irreversible.
Alfie's medical team have successfully argued that his life support should be turned off.
They lost a first round of cases in the High Court, Court of Appeal, Supreme Court and European Court of Human Rights.
In their written decision, they said there was "no reason for further delay".
Justice Hayden of the High Court ruled that whatever was causing Alfie's condition, the damage to the toddler's brain was so severe that it was in his best interest to be taken off life support.
But Mr Justice Hayden said flying Alfie to a foreign hospital would be wrong and pointless. The ruling means that a previous court's decision that Alfie be removed from his ventilator on Monday, April 23 remains in effect. The supreme court justices and European court of human rights' judges refused to intervene.
The parents made an application to appeal for a writ of habeas corpus, that requires a person having custody of the relevant party either to produce the body or to give good legal reason why the body should not be produced or released. It is a piece of common law which probably dates to Anglo-Saxon times.
After the Supreme Court's announcement, a legal representative said the family are to make another appeal to courts in Strasbourg to keep their son alive.
"Alfie looks like a normal baby", they said, "but the unanimous decision of doctors who have examined him and the scans of his brain is that nearly all of his brain is destroyed". There will be no further stay of the Court of Appeal's order.