That's according to audio of the call posted online.
The pair, claiming to be Queen Elizabeth the Second and her son, Prince Charles, got through to Kate's private nurse at King Edward the Seventh Hospital.
They had a brief conversation that included some details of her condition.
The hospital says it deeply regrets the incident, it takes patient confidentiality very seriously and is reviewing its telephone protocols, and the radio show has posted an apology for the call on its official Twitter feed.
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