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.
Arthur is also one of the duke's middle names, and was a middle name of King George VI - the Queen's father.
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