Jessica Bridge, the company's spokeswoman, said: "The support for Alice is starting to get silly now, but we've been left with no option but to cut the odds as punters relentlessly back the girl/Alice combination."
Two people have bet more than 2000 each on the baby being a girl, helping to shift the odds to 8/15 for a girl, with a boy now an 11/8 shot.
It would appear to be a case of the blind leading the blind - after all, the chances of a boy or a girl are 50/50, and there have been no reports suggesting the Duke and Duchess have chosen a name. However, punters were right about the name George last time, making it the overwhelming favourite before Prince George's birth, and that may have convinced people that someone, somewhere, knows something this time too.
Alice is such an overwhelming favourite that the next most fancied name, Elizabeth, has comparatively long odds of 6/1, as does Charlotte, with Victoria and Diana both at 14/1.
The odds on boys' names are even longer. Arthur and James, the two most fancied boys' names, both have odds of 20/1 at Ladbrokes.
Bookmakers are also taking bets on the day the baby will be born, and April 20 (UK time) emerged as the favourite in early betting.
Parking restrictions have been put in place outside the Lindo Wing of St Mary's Hospital in Paddington, in anticipation of the birth, although the restrictions will only be in place until April 30.
There is also the remote possibility that if the baby is overdue it could be born on May 7, the day of the General Election, for which bookies are offering odds of 20/1.
Top names for William and Kate's new arrival
•Alice (5/4)
•Elizabeth (6/1)
•Charlotte (6/1)
•Victoria (14/1)
•Diana (14/1)