The 46-year-old woman, who was found on Friday, is understood to have taken her own life.
In a statement, St James's Palace said: "The Duke and Duchess of Cambridge are deeply saddened to learn of the death of Jacintha Saldanha.
"Their Royal Highnesses 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 Scotland Yard spokesman said: "Police were called at approximately 9.35am this morning to a report of a woman found unconscious at an address in Weymouth Street, W1.
"London Ambulance Service attended and the woman was pronounced dead at the scene. Inquiries are continuing to establish the circumstances of the incident.
"The death is not being treated as suspicious at this stage."
The King Edward VII Hospital expressed very deep sadness and said that the nurse had worked there for more than four years.
"She was an excellent nurse and well respected and popular with all over her colleagues," a statement added.
"We can confirm that Jacintha was recently the victim of a hoax call to the hospital. The hospital has been supporting her at this difficult time."
It comes after two DJs from Australia's 2Day FM called the hospital on Tuesday and used fake accents - which they admitted were "terrible" - to pretend to be the Queen and Prince Charles.
We don't want to cause any trouble, we don't want to stress her (Kate) out because she is doing it tough. But I reckon we could maybe get her on the radio tonight ...
Michael Christian, 2Day FM
"If this has worked, it's the easiest prank call we have ever made," Michael Christian was heard to say on air to his colleague Mel Greig.
Staff fell for the hoax and were tricked into giving the pair an update on the pregnant Duchess.
When it was confirmed that Kate was stable and sleeping, Mel Greig replied: "Well, I'll just feed my little corgis then ... So when is a good time to come and visit her? Because I'm the Queen, so I'll need a lift down there ..."
Hospital chief executive John Lofthouse hit out at the prank later that day, branding it "deplorable".
He added: "Our nurses are caring, professional people trained to look after patients - not to cope with journalistic trickery of this sort."
Kate had been admitted to the hospital suffering from a form of extreme morning sickness, which can cause serious complications. She has since been discharged and is recovering at home.
Mr Lofthouse said there was no chance the Duchess could have received the call, but admitted it had technically been a breach of patient confidentiality - which he regretted.
"Having said that, the information which was inadvertently revealed is already in the public domain," he added.
St James's Palace further confirmed 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."
They were the worst accents ever and, when we made that phone call, we were sure 100 people at least before us would have tried the same thing.
Mel Greig, 2Day FM
The prank call was pre-recorded and vetted by lawyers before being broadcast to listeners in Sydney.
The DJs later apologised for their actions, as did their radio station, but have not yet spoken about news of the death.
The pair have since been subjected to a flood of abuse on social media websites and seem to have since closed their personal Facebook and Twitter pages.
The radio station continued to promote its prank call on air during what was the early hours of Saturday morning in Sydney.
It called it "the prank call the world is talking about" before playing clips of news programmes reporting on the original call, in the middle of a dance music programme.