Kate, 30, is being treated for very acute morning sickness – hyperemesis gravidarum – at a private London hospital.
A spokesman said the Duchess, who is thought to be less than 12 weeks pregnant, is likely to stay there for several days and "will require a period of rest thereafter".
The Royal baby will be born third in line to the throne, after Prince Charles and Prince William.
If it is a girl she will be the first to take advantage of new rules of succession, which are being drawn up and will mean that if she has younger brothers in the future, she will be able to succeed to the throne.
Speculation is rife over what the Duke and Duchess will name the child. B ookies are giving favourite names as Frances and John, both at 9/1, while Charles, Victoria and George are 10/1.