01 BST.
The statement from the Kensington Palace read, "Her Royal Highness and her child are both doing well".
The name of the baby will be announced in due course.
The Duchess was admitted to the private Lindo Wing at St. Mary's Hospital in Paddington, west London, in the morning hours of April 23.
The couple already has two children - a son, Prince George, aged 6, and a daughter, Princess Charlotte, aged 2.
The new prince is the Queen's sixth great-grandchild.
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