Ms Saldanha, a 46-year-old mother of two, was found unconscious near the King Edward VII Hospital, where the Duchess spent three nights earlier this week being treated for severe pregnancy sickness.
Paramedics tried to revive her but she was pronounced dead at the scene.
Police said the cause of her death was "unexplained" but they were not treating it as suspicious.
The Duke and Duchess of Cambridge said they were "deeply saddened" to be told of her death.
In a statement, the hospital said: “It is with very deep sadness that we confirm the tragic death of a member of our nursing staff, Jacintha Saldanha. Jacintha has worked at the King Edward VII Hospital for more than four years, she was an excellent nurse and a well respected and popular member of staff with all her colleagues.
“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.”
John Lofthouse, chief executive of the hospital, said: “Our thoughts and deepest sympathies at this time are with her family and friends. Everyone is shocked by the loss of a much-loved and valued colleague.”
Lord Glenarthur, chairman of the hospital, said: “This is a tragic event. Jacintha was a first class nurse who cared diligently for hundreds of patients during her time with us. She will be greatly missed.”
A St James's Palace spokesman said: "The Duke and Duchess of Cambridge are deeply saddened to learn of the death of Jacintha Saldanha.
|Guard: An officer guards the entrance to the block where the nurse's body was found|
"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."According to the Daily Mail, the circumstances in which she was found indicated she might have killed herself.
A spokesman for the Metropolitan Police said: "Police were called at approximately 9.25am on Friday, December 7, to a report of a woman found unconscious 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."
A spokesman for the hospital said a statement was expected shortly.
Presenters: Mel Greig (left) and Michael Christian (right) made the call about the Duchess - which has been directly linked to Jacintha's death by the hospital
The dead woman is reported to be a nurse who was working on reception who took the call at 5.30 on Tuesday morning from two presenters from the Sydney radio station 2Day FM, who pretended to be the Queen and the Prince of Wales.
The unsuspecting hospital worker put the call straight through to the Duchess's ward, where a nurse spent two minutes chatting about the Duchess's condition and treatment.
The hospital described the incident earlier this week as "deplorable".
Michael Christian, the DJ who pretended to be the Prince of Wales, apologised earlier this week but carried on tweeting about it and only this morning tweeted: "MORE on the #royalprank after 7.30 tonight."