The child, who was born in the Lindo Wing of St Mary's Hospital, central London, is fifth in line to the throne and the Queen's sixth great-grandchild.
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".
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.
Both were also called in for the arrival of Prince George in 2013 and Princess Charlotte in 2015.
The name of the new child has not yet been announced.
Favourite names at the bookmakers include Mary, Alice, Alexandra, Elizabeth and Victoria for a girl and Arthur, Albert, Frederick, James and Philip for a boy.
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