Doctors treating Charlotte Wyatt, the baby clinging to life in an oxygen box, have told her parents saying she has made remarkable progress.
Darren Wyatt, aged 33, originally from Tamworth, Staffordshire, and his wife, Debbie, 24, will take the letter to the Court of Appeal on Thursday to persuade judges to overturn the order not to resuscitate the 22-month-old girl if she stops breathing.
Doctors at St Mary's Hospital in Portsmouth won the legal right last October not to resuscitate Charlotte after arguing that her brain and other organs were so seriously damaged that she had "no feeling other than continuing pain".
Against the odds and medical opinion, she survived the winter and now responds to loud noise and bright images - and she even smiles.
She spends most of her time in an oxygen box, but is taken out to be cuddled by her parents when they visit.
Charlotte weighed only 1lb and measured only five inches long when she was born three months prematurely in October 2003. She has serious brain, lung and kidney damage.
Her survival confounded predictions that she would not last the winter because it was feared she was bound to succumb to a respiratory infection.
Mrs Wyatt, from Portsmouth, said: "When the court made that ruling they said they would not resuscitate her because of her quality of life. But now her quality of life has improved so we want them to lift the order.
"Charlotte can go outside for 40 minutes in the hospital grounds with an oxygen mask.
"She can see and hear, and she knows who Darren and I are."
At a hearing in April this year, Mr Justice Hedley was told by doctors that resuscitating her if she stopped breathing would be "pointless and possibly inhumane" because it would only prolong her suffering.
Mr Hedley renewed the court order allowing hospital doctors to let her die in peace if she stops breathing, saying: "I am quite clear that it would not be in Charlotte's best interests to die in the course of futile aggressive treatment."
The judge, in the High Court Family Division, stressed that the court order remained subject to review and that he would reconsider it in the light of any fresh evidence later in the year.
A letter, dated August 11, sent to Charlotte's parents said staff had been encouraged by her remarkable progress to date but insisted there was no change in her underlying condition.
A spokeswoman for Portsmouth Hospitals NHS Trust said: "This is part of the Trust's efforts to keep communication with the family about Charlotte's progress.
"As explained in the consultant's letter, Charlotte's underlying condition remains the same."
The couple, who are devout Christians, also have two sons, Daniel, three, and ninemonthold David and Mrs Wyatt is pregnant with their fourth child.
They will appeal on whether the court in April, this year, had a right to make a judgment in advance of a crisis.