The body of Jacintha Saldanha, who was working on the switchboard, was found this morning at an address yards away from King Edward VII Hospital, where Kate was being treated this week.
The Duke and Duchess of Cambridge issued a statement saying they are "deeply saddened" by the tragedy and said: "Their thoughts and prayers are with Jacintha Saldanha's family, friends and colleagues at this very sad time."
The King Edward VII hospital, who were not believed to be disciplining the mother of two over the incident, said it announced her death with "very deep sadness".
"Jacintha has worked at the King Edward VII Hospital for more than four years. She was an excellent nurse and well respected and popular with all over her colleagues," the statement was quoted as saying by the Daily Mail.
Nurse Saldanha answered the hoax call at 5.30am on Wednesday morning, and was helping out on reception at the time of the prank.
Giggling DJs Mel Greig and Michael Christian were pretending to be the Queen and Prince Charles and asked her if they could be put through to Catherine.
Jacintha connected them to another nurse who gave details of Kate's condition, who was suffering from acute morning sickness at the London hospital.
The presenters, from the radio station, remarked during their show how their efforts were the "easiest prank call ever made", as they put on mock British accents they later described as "terrible".
The emergency services were called and two ambulances were dispatched to the central London residence today.
But paramedics were unable to revive her and she was pronounced dead at the scene.
Her family, husband and two children have been informed.
The hospital said this afternoon: "It is with very deep sadness that we confirm the tragic death of a member of our nursing staff, Jacintha Saldanha."
"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," the hospital said.
Officers from Scotland Yard have launched an investigation and are treating the death as "unexplained".
The exact cause of death remained unclear. However, one source indicated that the woman appeared to have killed herself.