William Hill is offering odds of 10/11 that the baby will be a boy and the same price for a girl.
The favourite names are Frances and John, both with odds of 9/1, with Charles, Victoria and George all 10/1.
There are also odds of 12/1 the newborn will be called Diana, Phillip, Richard or Spencer.
"We fully expect that the betting public will get involved in speculating what the baby will be called," said William Hill spokesman Joe Crilly.
"When William and Katherine got married we were bowled over by a million pounds' worth of bets and we predict similar levels of excitement before the birth of their first child."
Odds on hair colour are 6/4 brown, 2/1 blonde, 5/2 black and 8/1 ginger.