Two presenters on Sydney-based radio station 2Day FM are reported to have convinced staff at King Edward VII hospital in London that they were the Royals and were connected to the duchess's private nurse yesterday.
The presenters posed as the Queen and Prince Charles got details of Kate's health condition from a nurse.
The Australian 2DayFM placed the recording of the conversation on its website.
The leading private hospital treating the 30-year-old Duchess of Cambridge said it "deeply regrets" giving out information about her condition to hoax callers.
The hospital said confidentiality was taken seriously and telephone protocols are under review, the BBC reported.
A spokesman for the hospital said: "This call was transferred through to a ward and a short conversation was held with one of the nursing staff. King Edward VII's Hospital deeply regrets this incident."
John Lofthouse, chief executive at King Edward VII's Hospital, said: "This was a foolish prank call that we all deplore."
A spokesman for William and Catherine said he would be making no comment.
The duchess, who was admitted to the hospital on Monday, is thought to be less than 12 weeks pregnant.
It has been confirmed she is being treated for acute morning sickness, or hyperemesis gravidarum.
Her husband Prince William, who spent several hours visiting her on Monday, spent the day with her again yesterday.
News of Kate's pregnancy has ended feverish speculation about a new royal heir that began immediately after her marriage to Prince William in April 2011.
Meanwhile, a St James's Palace spokesman said: "The Duchess of Cambridge is continuing to feel better. She and the Duke are immensely grateful for the good wishes they have received. She will remain in hospital at present."