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”.
Kensington Palace said the name of the baby would be announced in due course and a formal notice of the birth would be posted on the forecourt of Buckingham Palace.
Favourite names at the bookmakers include Arthur, Albert, Frederick, James and Philip.
Prime Minister Theresa May tweeted her “warmest congratulations” to William and Catherine.
She said: “I wish them great happiness for the future.
Mayor of London Sadiq Khan tweeted: “On behalf of Londoners, I want to send our warmest congratulations to the Duke & Duchess of Cambridge on the birth of their baby boy.”
The duchess was admitted to hospital shortly before 6:00 on Monday with the announcement of the birth coming shortly after 13:00.
Campaign corks popped and cheers rang out as the news the duchess had given birth to a boy ran through the crowd.
Those who had been camped out for days led the chants: “It’s a boy. It’s a boy.”
Others began singing: “Congratulations, and celebrations. It’s a boy born on St George’s Day.”
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 new prince, born on St George’s Day, shares a birthday with with Lady Gabriella Windsor – the daughter of Prince and Princess Michael of Kent, who was born at the Lindo Wing on 23 April 1981.
Image copyrightPA
Image copyrightPAWill the new prince be king?
To become King or Queen as the third-born royal child is rare – and has yet to happen within the current House of Windsor.
But the third child of George III and Queen Charlotte, William IV, took on the task and ruled from 1830 to 1837.
The Hanoverian king acceded to the throne aged 64 when his older brother, George IV, died without an heir.
He became next in line when he was 62 and his other older brother, Frederick, Duke of York, died.
