The French embassy to London tweeted: "We're delighted that you've been given such a lovely French name!"
While parents in the United Kingdom legally have 42 days to officially register a child's birth, William and Kate are expected to register their son's arrival this week, judging by how long it took them for George and Charlotte.
Royal commentator Victoria Howard told News Corp the couple would have run the name past the Queen.
Pottermore, author JK Rowling's publishing company, instead pointed out that Louis, Arthur and Charles were all members of the Weasley family from the bestselling Harry Potter books. "You'll find out soon enough", he said with a smile.
Joking about the baby prince's name after Australian High Commissioner Alexander Downer asked him if he's considered the name Alexander for the little one, William replied: "Well, it's amusing you should say that".
The baby was born weighing eight pounds and seven ounces (3.8 kilogrammes) at 11:01 am (1001 GMT) on Monday with William, the Duke of Cambridge, present for the birth.
Louis is fifth in line to the throne after his grandfather Prince Charles, his father and his older siblings.
His birth was also a significant moment for Charlotte, the first princess who will not be overtaken in the line of succession by her younger brother. It also pays tribute to the Prince of Wales's great-uncle, Earl Mountbatten, who was murdered by the IRA.
Known to Charles as "Uncle Dickie", Mountbatten was the last British Viceroy of India before independence in 1947.
His boat was blown up on a fishing trip off the coast of County Sligo in the Republic of Ireland.
"The Queen would obviously be consulted and, while she might not necessarily intervene and prevent a name from being used, she would certainly be aware of the name, long before any public announcement".