The couple’s second son was born at 11:01 BST on Monday at the Lindo Wing of St Mary’s Hospital in London, weighing 8lb 7oz.
A tweet from Kensington Palace confirmed the baby would be known as His Royal Highness Prince Louis of Cambridge.
The new prince is fifth in line to the throne.
Louis is one of the middle names of Prince William and of the new prince’s brother, George.
It was also the first name of Lord Mountbatten his great-great-uncle – who was killed by the IRA in 1979.
Arthur is also one of the duke’s middle names, and was a middle name of King George VI – the Queen’s father.
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. pic.twitter.com/4DUwsLv5JQ
— Kensington Palace (@KensingtonRoyal) April 27, 2018
Bookmakers had Arthur as the favourite name for the prince, followed by James and Philip.
On Thursday evening, Prince William told that they were still deciding on a name, saying: “I’m working on that.”
Earlier in the week, he said Catherine 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.”>