She had been the victim of two Australian radio DJs who impersonated Queen Elizabeth and Prince Charles to obtain a condition update on Kate, who was suffering severe morning sickness at the private hospital.
"The Duke and Duchess of Cambridge are deeply saddened to learn of the death of Jacintha Saldanha," the palace stated.
"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."
The hospital said in a statement: "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.
"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."
The two presenters, Mel Greig and Michael Christian from 2Day FM, 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 presenters later apologised for their actions, as did their radio station.
The DJs made their call at around 5.30am on Tuesday and are thought to have been put through by Ms Saldanha (46).
Another nurse 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."
The nurse added: "She hasn't had any retching with me since I've been on duty and she has been sleeping on and off. I think it's difficult sleeping in a strange bed as well."
The prank call was pre-recorded and vetted by lawyers before being broadcast.
In their initial apology the two presenters said: "We were very surprised that our call was put through. We thought we'd be hung up on as soon as they heard our terrible accents. We're very sorry if we've caused any issues and we're glad to hear that Kate is doing well."