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 said the baby would be known as His Royal Highness Prince Louis of Cambridge (pronounced Lou-ee, not Lou-iss).
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-great uncle - who was killed by the IRA in 1979.
Lord Mountbatten had a strong influence in the upbringing of great-nephew Prince Charles and was described by the latter as "the grandfather I never had"
Arthur is also one of the duke's middle names, and was a middle name of King George VI - the Queen's father.