The death, which police said they were treating as unexplained, comes days after the King Edward VII hospital apologised after falling for the call from an Australian radio station and relaying details about Kate's condition.
"It is with very deep sadness that we confirm the tragic death of a member of our nursing staff, Jacintha Saldanha," the hospital said in a statement. (Read: full statement)
"We can confirm that Jacintha was recently the victim of a hoax call to the hospital. The hospital had been supporting her," the statement said.
Police said they had been called at 9:35 am (15:05 IST) about a woman found unconscious at an address near the hospital. The woman was pronounced dead at the scene by ambulance staff.
The announcement on Monday that Kate was pregnant with a future British king or queen sparked a media frenzy and generated worldwide interest.
William and Kate, who left the hospital on Thursday, said they were "deeply saddened" by the death.
"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 statement from William's office said.
Two presenters from Australia's 2Day radio station called the hospital early on Tuesday British time, pretending to be William's grandmother Queen Elizabeth and his father, the heir-to-the throne Prince Charles.
Presenters Michael Christian and Mel Greig were put through to the ward where Kate was being treated and were given details about how she was faring.
The hospital issued a statement after the prank was revealed saying it deeply regretted the incident and said it was reviewing its "telephone protocols".
The prank call and its tragic aftermath come as Britain's own scandal-hungry press scrambles to agree a new system of self-regulation following a damning inquiry into their reporting practices.
A recording of the call was widely available on the Internet and many newspapers printed a transcript of the call.
The Australian radio station had previously apologised for the call, saying it was done with the "best intentions".
But some people leaving comments on the station's website had been unimpressed. "You have probably cost a young nurse her job ... Never mind that you violated a woman's privacy. Never mind, it was all done for fun," one said.
Facebook tribute pages swiftly set up after the nurse's death attracted messages of sympathy, some calling for the radio station to pay compensation to her family and others for the presenters to resign.
Saldanha's body was removed from the redbrick, five-storey building where it was found, and transferred to a small private ambulance, shortly after the hospital confirmed her death, a Reuters reporter at the scene said.
Detectives and forensics staff had earlier been seen entering and leaving the building, which contains residential apartments and administrative offices for the hospital.