Justin Welby tweeted: "Congratulations to The Duke and Duchess of Cambridge on the safe delivery of their baby, a brother for Prince George and Princess Charlotte. May God bless them and all of their children with love, happiness and health."
The birth has left the royal family and the Middletons "delighted", Kensington Palace has said.
Mother and baby are "both doing well" after the infant prince was born at 11.01am, weighing 8lb 7oz.
The baby's sex was a surprise to William and Kate, who chose not to find out what they were having.
Prime Minister Theresa May led the messages of support for the couple, saying: "My warmest congratulations to the Duke and Duchess of Cambridge on the birth of their baby boy.
"I wish them great happiness for the future."
The new prince of Cambridge is fifth in line to the throne and a younger sibling to Prince George and Princess Charlotte.
He was born more than five hours after the duchess was driven to the private maternity wing of St Mary's Hospital in Paddington, central London. She arrived before 6am, in the early stages of labour, with the duke by her side.
A huge media presence soon gathered outside the private Lindo Wing in anticipation of the birth, the same unit where George and Charlotte were born and their father the duke.
The news of the birth was announced by Kensington Palace a few minutes after 1pm in a statement, and later there was a more formal proclamation when brief details of the delivery were printed on a bulletin placed on an easel erected in the forecourt of Buckingham Palace.
Kensington Palace said the baby's name would be "announced in due course", but Arthur is already the favourite with punters, according to the bookies.
Kensington Palace said in its statement: "Her Royal Highness the Duchess of Cambridge was safely delivered of a son at 1101hrs.
"The baby weighs 8lb 7oz. The Duke of Cambridge was present for the birth.
"The Queen, the Duke of Edinburgh, the Prince of Wales, the Duchess of Cornwall, Prince Harry and members of both families have been informed and are delighted with the news.
"Her Royal Highness and her child are both doing well."
The new royal baby has been born on patriotic St George's Day - which celebrates the patron saint of England, and arrived just two days after the Queen's 92nd birthday.
At the exclusive Lindo Wing, the duchess's labour was overseen by consultant obstetrician Guy Thorpe-Beeston, surgeon-gynaecologist to the household, and consultant gynaecologist Alan Farthing, the Queen's surgeon-gynaecologist - part of the trusted team who delivered George and Charlotte.
They were joined by two other senior figures, Dr Sunit Godambe, a consultant neonatologist at Imperial College NHS Trust, which runs St Mary's hospital, he was also involved in Kate's previous births.
Professor Huw Thomas, physician to the Queen and head of the medical household, was the final member of the team, and he also cared for Kate when the duchess's daughter was born.
In addition, a group of top medics was on stand-by in case of an emergency.
The duchess will be expected to head home to the sanctuary of Kensington Palace in a few hours, just as she did with Charlotte in 2015 - but not before introducing the new royal baby to the world on the steps of the Lindo Wing.
George and Charlotte might even visit the hospital to see their mother and baby sibling, just as George did when his younger sister was born three years ago.
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