The hospital "had been supporting her throughout this difficult time," it said.
A St. James' Palace spokesman said: "The Duke and Duchess of Cambridge are deeply saddened to learn of 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."
Separately, a palace spokesman told CNN: "At no point did the palace complain to the hospital about the incident. On the contrary, we offered our full and heartfelt support to the nurses involved and hospital staff at all times."
The hospital said Wednesday that it deeply regretted the call had been put through.