Jacintha Saldanha, 46, was pronounced dead yesterday morning at an address near the King Edward VII's Hospital in central London where the pregnant duchess had been treated for a severe form of morning sickness.
The nurse, reportedly a mother-of-two, was the victim of two Australian radio DJs who impersonated the Queen and the Prince of Wales.
She answered their call and, believing the 2Day FM presenters were members of the royal family, put them through to another nurse who described Kate's condition in detail.
In a statement, Ms Saldanha's family said they were deeply saddened by the death and asked for privacy.
"We as a family are deeply saddened by the loss of our beloved Jacintha."
The company which runs the radio station said the hosts would not be returning to their show until further notice by mutual consent.
In a statement Southern Cross Austereo said: "Southern Cross Austereo [SCA] and 2Day FM are deeply saddened by the tragic news of the death of nurse Jacintha Saldanha from King Edward VII's Hospital and we extend our deepest sympathies to her family and all that have been affected by this situation around the world.
"Chief executive officer Rhys Holleran has spoken with the presenters. They are both deeply shocked and at this time we have agreed they will not comment about the circumstances.
"SCA and the hosts have decided they will not return to their radio show until further notice out of respect for what can only be described as a tragedy."
It is not known how much contact, if any, Kate had with the nurse, but 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's Hospital, and their thoughts and prayers are with Jacintha Saldanha's family, friends and colleagues at this very sad time."
The hospital said in a statement: "We can confirm the tragic death of a member of our nursing staff, Jacintha Saldanha.
"Jacintha has worked at the King Edward VII's 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 Jacintha was recently the victim of a hoax call to the hospital. The hospital had been supporting her at this difficult time."
Hospital chief executive John Lofthouse 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."
A spokesman for St James's Palace stressed it had not complained to the hospital about the hoax call.
He added: "On the contrary, we offered our full and heartfelt support to the nurses involved and hospital staff at all times."
Scotland Yard said officers were called at around 9.35am yesterday. A woman was reported to have been found unconscious at an address in Weymouth Street in central London. It said: "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 presenters, Mel Greig and Michael Christian from 2Day FM, remarked during their show that their efforts were the easiest prank call ever made, as they put on mock British accents they later described as terrible.
The DJs, who later apologised, made their call at around 5.30am on Tuesday and were put through by Ms Saldanha. Another nurse looking after Kate told them: "She's sleeping at the moment and she has had an uneventful night and sleep is good for her.
"She's been given some fluids to rehydrate her because she was quite dehydrated when she came in but she's stable at the moment."