The couple’s third child, who was born on Monday April 23 at 11.01am, weighing 8lb 7oz, will be known as Prince Louis, Kensington Palace announced after a four-day wait.
In a short statement the Palace said: “The Duke and Duchess of Cambridge are delighted to announce that they have named their son Louis Arthur Charles.
“The baby will be known as His Royal Highness Prince Louis of Cambridge.”
The baby is fifth in line to the throne and the Queen and the Duke of Edinburgh’s sixth great-grandchild.
William and Kate’s choice came as a surprise because the name is already one of Prince George’s middle names.
It pays tribute to the Prince of Wales’s great-uncle, Earl Mountbatten, who was murdered by the IRA.
Bookmakers had Arthur as the favourite for some time, with James and Philip also popular.
Alexander became a later front-runner after William remarked on the name at a church service, as did Albert as the wait for the baby’s name stretched on.
The middle name Charles is a tribute to the Duke’s father, the Prince of Wales.
The Duke and Duchess will have informed the Queen before asking the Palace to announce their son’s name.
On Thursday evening at an event with Prince Harry, William revealed that the couple were still deciding, telling reporters who asked about the name: “I’m working on that.”
He also took a break from caring for the new prince to carry out royal duties on Wednesday when he joined Harry and Meghan Markle at an Anzac Day service in London.
The Duke told guests that Kate and the baby were “in good form, luckily”, and added: “Sleeping’s going reasonably well so far, so he’s behaving himself, which is good news.”
He hinted at his name preferences when he chatted to the Australian High Commissioner, Alexander Downer, at Westminster Abbey, who asked William: “Have you thought of the name Alexander for him?”
The Duke replied: “Well, it’s funny you should say that. It’s a good name.”
And the Duke also remarked on how Sir Jerry Mateparae, New Zealand High Commissioner to the UK, had a “strong name”.
William and Kate’s third child is a younger brother to Prince George and Princess Charlotte.
Four-year-old George was christened George Alexander Louis in 2013, while Charlotte, who turns three next week, was named Charlotte Elizabeth Diana in 2015.
Their names were announced when they were two days old.
William joked on Monday that he had “thrice the worry” now as he took the prince home to Kensington Palace, after the baby’s debut on the steps of the exclusive Lindo Wing maternity unit in London.
The Duchess is on maternity leave but is expected to attend Harry and Ms Markle’s high-profile wedding in Windsor on May 19.