Harry said the name would be revealed later this week.
Royal names in Britain are normally chosen from a relatively restricted pool of those used by past royals.
This presents Harry and Meghan -- a mixed-race former US TV actress -- with the difficult choice of whether to follow royal tradition or choose a more unusual name.
Her Royal Highness The Duchess of Sussex was safely delivered of a son at 0526hrs.
— The Royal Family (@RoyalFamily) May 6, 2019
The baby weighs 7lbs 3oz and The Duke of Sussex was present for the birth.
Read the full announcement here: https://t.co/RCUFjQG8pe
However, since their baby is only seventh in line to the throne and therefore unlikely to ever become king, the couple may have more freedom to make their own choice.
When Meghan ran a lifestyle blog called The Tig before her wedding last year, she wrote that one of her favourite baby names was Grey, which would work both a boy and a girl.
It’s a boy! Prince Harry, Meghan welcome their first child
That is unlikely to go down well in Britain and the same would go for very American-sounding names such as Bradley, which is seen as highly improbable by the bookmakers.
Slightly higher in the odds ranking is "Spencer" -- a popular name in the United States that is also the maiden name of Harry's late beloved mother Princess Diana.
Rupert Adams from bookmaker William Hill said he had seen "a huge uplift" in betting on the name since the birth.
"Arthur is the new favourite but we could see a lot of changes in the odds in the coming hours or even days," Adams said.
At Paddy Power, the joint favourites were "Alexander" and "James", although the odds of "Oliver" were also narrowing.
Arthur, which dates back to the legend of King Arthur, is a popular middle name in the royal family -- held by Prince Charles, Prince William and William's son Louis.
Albert was the name of Queen Victoria's husband who insisted that it be used as a middle name by her descendants, while Philip would be a lasting tribute to Queen Elizabeth's husband -- the Duke of Edinburgh.
During the early days of Meghan's pregnancy, her father-in-law Prince Charles joked that the couple had received lots of suggestions during a tour of Australia.
"Just between us, I suspect that Kylie and Shane may possibly make the shortlist," he said in a speech at Australia House in London.
"But, ladies and gentlemen, I would not hold your breath for Edna or Les," he quipped.
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