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http://www.bbc.co.uk/news/uk-20645838
The Duke and Duchess of Cambridge said in a statement they were "deeply saddened" by the death of the nurse, named as Jacintha Saldanha.
King Edward VII hospital paid tribute to "a first class nurse who cared diligently for hundreds of patients".
Details of the pregnant duchess's medical condition were unwittingly revealed to two Australian DJs.
Mel Greig and Michael Christian, from Sydney radio station 2Day FM posed as the Queen and Prince Charles in a call early on Tuesday morning.
Rhys Holleran, chief executive of SCA, the company which owns the station, said in a statement the pair will not return to their show until further notice out of respect for Mrs Saldanha.
The duchess had been admitted on Monday for acute morning sickness, and was discharged on Thursday.
'Excellent nurse'
BBC royal correspondent Peter Hunt said he understood Mrs Saldanha - who was staying in hospital accommodation close to King Edward VII hospital - was the person who answered the call from the Australian DJs and was not the nurse who discussed the duchess's medical condition.
Mel Greig and Michael Christian had said they were "very sorry if we've caused any issues"
Mrs Saldanha, a duty nurse who was married with two children, answered the telephone because it was 05:30 GMT and there was no receptionist on duty.
The BBC understands Mrs Saldanha had not been suspended or disciplined by the hospital.
The BBC's Nicholas Witchell said it had been suggested to him that she had felt "very lonely and confused" as a result of what had happened.
The St James's Palace statement said the duke and duchess "were looked after so wonderfully well at all times by everybody at King Edward VII Hospital, and their thoughts and prayers are with Jacintha Saldanha's family, friends and colleagues at this very sad time".
A palace spokesman later added that "at no point did the palace complain to the hospital about the incident".
"On the contrary, we offered our full and heartfelt support to the nurses involved and hospital staff at all times."
The duchess left the hospital on Thursday
In a statement outside the hospital, King Edward VII chief executive John Lofthouse said: "We can confirm that Jacintha was recently the victim of a hoax call to the hospital.
"The hospital had been supporting her throughout this difficult time."
He said Mrs Saldanha, who had worked at the hospital for more than four years, "was an excellent nurse and well-respected and popular with all of her colleagues".
"Everyone is shocked by the loss of a much loved and valued colleague," he added.
Royal College of Nursing chief executive Dr Peter Carter, meanwhile, said it was "deeply saddening that a simple human error due to a cruel hoax could lead to the death of a dedicated and caring member of the nursing profession".
Scotland Yard said officers were called at 09:35 GMT on Friday after reports of a woman found unconscious at an address in Weymouth Street, central London. She was pronounced dead at the scene.
Police said the death was not being treated as suspicious.
In a statement, Mrs Saldanha's family said they were "deeply saddened by the loss of our beloved Jacintha". They have requested privacy from the media.
'Very sorry'
The prank call was pre-recorded before it was assessed by lawyers and broadcast on 2Day FM.
Speaking on their show, Mel Greig and Michael Christian, called it the "easiest prank call ever made" and described their mock British accents as "terrible".
In the call, another nurse was tricked into revealing specific confidential information about the duchess's medical condition.
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We're going to have a long and careful think about what, if anything, we do?
King Edward VII chief executive John Lofthouse, speaking on Tuesday
The DJs later apologised saying they were "very surprised that our call was put through".
"We thought we'd be hung up on as soon as they heard our terrible accents.
"We're very sorry if we've caused any issues and we're glad to hear that Kate is doing well."
The Twitter accounts of both presenters have been deleted and all references to the prank call recording removed from the 2Day FM website.
SCA said on its Facebook page the company was "deeply saddened by the tragic news" and had extended "our deepest sympathies to her family and all that have been affected."
On Tuesday, hospital chief executive John Lofthouse said he had "received advice that what the Australian broadcasters did may well have broken the law".
But he added: "On the other hand they've apologised for it so we're going to have a long and careful think about what, if anything, we do."
The Australian Communications and Media Authority (Acma) has said it had received complaints about the call.
Radio station 2Day FM has previously been in trouble with Acma for incidents including one when a 14-year-old girl revealed on air that she had been raped.
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