The former Kate Middleton is due to give birth to her first baby in July, and her health is improving after a bout of severe morning sickness, palace officials said on Monday.
The child of Kate and Prince William will be third in line to the British throne.
The 31-year-old Duchess of Cambridge, as she is formally known, spent several days in hospital last month being treated for acute morning sickness.
She has since resumed public appearances, and looked healthy at the unveiling of an official portrait last week.
The palace said in a statement that the duchess' condition continues to improve.
It said the two "are delighted to confirm they are expecting a baby in July".
The announcement by the couple's St James's Palace office lays to rest speculation that the duchess could be having twins.
The palace would not comment on whether the baby is a boy or a girl.
The British Government is changing centuries-old succession laws to ensure that a royal daughter will have the same claim to the throne as a son.
Bookmaker William Hill released odds on the name of the royal baby, with favourites including George, Victoria and Diana, all at 10-1.