The princess – who is fourth in line to the throne – was “safely delivered” at 08:34 BST, the palace statement said.
The Duke of Cambridge, who was present for the birth of the 8lbs 3oz (3.7kg) baby girl, brought his 21-month-old son Prince George to visit his sister at St Mary’s Hospital, west London.
A large number of journalists, photographers and broadcasters had gathered after the announcement that Catherine was in labour.
Following the announcement of the birth, “We welcome with humble duty the new baby of their Royal Highnesses the Duke and Duchess of Cambridge.
She will be the first to take the title of Princess for 25 years.The new royal will not be overtaken in the line of succession by any future younger brothers.
Under new rules which came into force in March, male bias was removed from the succession rules. And the princess will also be able to marry a Roman Catholic without losing her place in line to the throne.