The 20-year-old woman, an accident victim whose baby was still- born, has already been given life- saving blood and plasma transfusions against her wishes.
A High Court judge ruled last week that doctors can legally give the transfusions even though the woman objected because of her religious beliefs - the Jehovah's Witnesses are against the use of artificial means to preserve life.
But Lord Donaldson, Master of the Rolls, and Lords Justices Butler-Sloss and Staughton have been asked to overturn that decision by ruling that the woman's wishes must come first.
The Official Solicitor, David Venables, who represents those who cannot act for themselves, has taken up the woman's case because she is now unconscious.
But her appeal is being opposed by her father and by the West Midlands health authorities responsible for treating her.
The woman, 'T', was injured in a road crash on 1 July. She was admitted to hospital three days later with suspected pleurisy or pneumonia. She was 34 weeks pregnant at the time and later gave birth by Caesarean section. The child was still-born and the mother needed blood transfusions.
James Munby QC, for the Official Solicitor, told the appeal judges that T was heavily sedated, virtually unconscious and in a critical condition. She might need an operation for an abcess, requiring further transfusions. 'This appeal raises for the first time in this country the question whether, and if so in what circumstances, an adult patient is entitled to refuse medical treatment in circumstances whereby that refusal may bring about her death,' he said.
The question of a 'constitutional right to die' had been before courts in Canada and the US. While it had been aired in the English courts in respect of children, this was the first time it had been raised in relation to an adult.
Two years ago courts ruled that the two-year-old daughter of Jehovah's Witnesses should have a transfusion. Only two weeks ago, the Court of Appeal ruled that young people aged 16 to 18, normally considered as adults when it comes to medical care, had no legal right to refuse treatment.
Mr Munby said the legal argument centred on whether the woman was capable of giving a legally valid refusal to receive transfusions or whether there was some feature of the case which justified ignoring her wishes.
He cited a famous judicial statement that every human being of adult years and of sound mind had the right to decide what should be done with his or her body.
The hearing continues today.