The odds for Grace, which was 33/1 in January, have been slashed to to 8/1.
Punters are confident the seventh in line to the throne will be a girl at 4/7, with three in every four bets on the gender favouring that outcome.
Diana (6/1) has long been a favourite name, which would pay tribute to Prince Harry’s late mother, Diana, Princess of Wales.
But both Diana and Grace sit behind the name Elizabeth, which takes the top spot at 5/1.
Alex Apati of Ladbrokes said: "Punters a convinced Harry and Meghan are having a girl and all the money right now is going on 'Grace' in the baby name betting."
The top five girls names are Elizabeth, Diana, Grace, Victoria and Alice.
The top five boys names are Albert (12/1), Philip, Arthur, James, and Alexander.
The Duke and Duchess of Sussex's child is due any day now, with Meghan on maternity leave from official royal duties.