The baby will take his or her place in the line of succession after grandfather Prince Charles, father William and elder siblings George and Charlotte.
Kate is widely expected to go into labor this month, just a few weeks before the marriage of William's brother Prince Harry and actress Meghan Markle on May 19.
The most popular name is Mary at 3/1, followed by Alice on 6/1 and Victoria on 8/1, William Hill said. For a boy, the leading bets are Albert, Arthur and Fred or Frederick, all on 12/1.
(Reporting by Costas Pitas; editing by Stephen Addison)