Kate's due date is the second half of this month but officials said the royal couple don't know the sex of their baby.
The Duke and Duchess were said to be 'hugely grateful' for the warm wishes they've received.
St Mary's Lindo Wing in Paddington, west London, describes itself as having offered 'leading private obstetric and neonatal care for 60 years'. It's based within an NHS hospital with facilities for premature babies, and access to hospital specialists.
It costs more than £5,000 for the first 24 hours of a delivery with an extra £1,000 for each additional night.
It has also seen the births of the children of Princess Anne, the Duchess of Gloucester and Princess Michael of Kent.
Prince George was delivered there at on 22 July 2013, weighing 8lb 6oz (3.8kg).
Hundreds of reporters, photographers and TV camera crews from across the world gathered outside ahead of his birth.
But BBC royal correspondent Nicholas Witchell said Kensington Palace is hoping the media will not start to turn up this time until it is confirmed that the duchess has been admitted to give birth.
He said: "I think on this occasion the police will move on any reporter, photographer, camera crews and people such as us, who attempt to linger there before it is confirmed that she has gone into hospital to give birth.
Meanwhile, it looks like Prince William could face a two-hour hospital dash from Norwich to London when his wife goes into labour.
The Duke is dividing his time between Norwich, Cambridge, and Staverton, Gloucestershire, during the final stages of his air ambulance pilot training and won't have a helicopter on standby to whisk him to London when the Duchess is admitted to St Mary's Hospital in Paddington.
Norwich is 118 miles from St Mary's and Staverton is 96 miles away.
To drive from Norwich to Paddington normally takes almost three hours, but the Duke's aides say he will be 'no more than two hours' drive away'.
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