A couple who want to take their sick baby son to the USA for treatment hope to mount a fight in the Supreme Court.
Chris Gard and Connie Yates want nine-month-old Charlie Gard, who suffers from a rare genetic condition and has brain damage, to undergo a therapy trial.
Specialists at Great Ormond Street Hospital in London, where Charlie is being cared for, say therapy proposed by a doctor in America is experimental and will not help.
They say life support treatment should stop.
A High Court judge last month ruled against a trip to America and in favour of Great Ormond Street doctors.
Mr Justice Francis concluded that life support treatment should end and said Charlie should be allowed to die with dignity
Three Court of Appeal judges on Thursday upheld that ruling.
Lawyers representing the couple say they will now ask Supreme Court justices to analyse the case.
Bosses at Great Ormond Street say doctors will continue providing life support treatment to Charlie over the next few days while applications are made.
A barrister representing the hospital told appeal judges how doctors felt they were reaching the stage where continuing to provide life support treatment would not be ethical.
The couple left the appeal court without comment.
Charlie's mother was clutching a law book called The Family Court Practice 2017 in her hand as she walked away.