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, has left St Mary's Hospital in London to "pick up" his 21-month-old son Prince George.
He also told the waiting crowds the couple were "very happy".
Both Catherine and her daughter are "doing well", Kensington Palace said.
The name of the baby will be announced in due course.
The Prince of Wales, who had earlier said he was hoping for a grand-daughter, and the Duchess of Cornwall said they were "absolutely delighted".
(BBC News)