2DayFM hosts Michael 'MC' Christian and Mel Greig phoned London's King Edward VII Hospital, where the duchess is currently being treated, and managed to get a member of hospital staff to give them an update on her medical condition by impersonating Queen Elizabeth II and Prince Charles.
"There is a lot of anger about this. Prince William was said to be furious when he found out," a palace source told The Mirror. "Kate is vulnerable and weak at the moment, and these people think it's funny to make a joke out of it?"
The duchess was admitted to hospital on Monday (December 3), as news broke that she is pregnant with the couple's first child. She is being treated for hyperemesis gravidarum, which is an acute form of morning sickness.
Christian and Greig have since apologised for the call, which was pre-recorded prior to transmission.
[Prince William leaving the King Edward VII Hospital earlier this week]
"We were very surprised that our call was put through," they said. "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."
2DayFM added: "[The station sincerely apologises for any inconvenience caused by the inquiries to Kate's hospital. The radio segment was done with lighthearted intentions.
"We wish Kate and her family all the best and we're glad to hear she's doing well."
The hospital has launched an investigation into its telephone procedures, and has described 2DayFM's stunt as "a foolish prank call that we all deplore".
Photo gallery - Catherine, Duchess of Cambridge: