The new arrival, who is the third child of Catherine and the Duke of Cambridge, was born at 11:01 BST weighing 8lbs 7oz.
A Kensington Palace statement said Catherine and her new son are "both doing well".
William was present for the birth, the palace added.
The statement added that members of both families had been informed and were "delighted with the news".
Prime Minister Theresa May tweeted her "warmest congratulations" to William and Catherine.
She said: "I wish them great happiness for the future."
The duchess was admitted to hospital shortly before 6:00 on Monday with the announcement of the birth coming shortly after 13:00.
Senior royal doctors consultant obstetrician Guy Thorpe-Beeston and consultant gynaecologist Alan Farthing oversaw the birth.