The third in line to the throne, who was born on Monday at 16:24 BST, will be known as His Royal Highness Prince George of Cambridge.
The Queen was told of Prince George's name when she met him for the first time on Wednesday.
The family are at the Middleton home in Bucklebury, Berkshire, after spending a night at Kensington Palace.
George was the bookmakers' favourite for the first name of the prince.
The duke and duchess had been expected to pick a traditional royal name for their son, who was born at St Mary's Hospital in London, weighing 8lbs 6oz.
Royal infants usually have historical names which are passed down through the generations.
There have been six King Georges up to now, most recently the Queen's father, although his first name was Albert and he was known to his family as Bertie.
The name Louis is Prince William's fourth name and is likely to be a tribute to Lord Louis Mountbatten, the Duke of Edinburgh's uncle and the last British Viceroy of India before independence in 1947.
"It is a product to help breast-feeding mothers," an entrepreneurial woman replies when asked what is in the boxes. "A food supplement. We want to get some to Kate but she is not here is she?"
Indeed the Duke and Duchess of Cambridge left Nottingham cottage in the grounds of Kensington Palace in the early afternoon and most passers-by are unaware that this week's most famous new parents have named their son.
Waiting reporters reveal the name to some of the enquiring public. Most are satisfied with the choice. "Not surprising," say others.
Only one crinkles up his nose when he hears the three names.
For a few minutes there is a buzz around the camera crews as broadcasters analyse the significance of George Alexander Louis and then the park returns to normal as the cyclists and runners drift by, oblivious that His Royal Highness Prince George of Cambridge, their future king, has been named.
Prince Charles was immensely fond of his great-uncle Lord Mountbatten. Known by the Royal Family as Uncle Dickie, he was assassinated by the IRA in August 1979.
The prince's other middle name - Alexander - is one that three medieval Scottish kings have had, and was also the name of the famous 4th Century ruler Alexander the Great. The Queen's middle name is Alexandra.
BBC royal correspondent Nicholas Witchell said George will be seen as a tribute to the Queen's father but the couple are said to genuinely like the name themselves.
Alexander was a favourite of Catherine's, he added.
The BBC's Peter Hunt said Prince George does not have to be King George if he accedes to the throne - he can choose another name, as the Queen's father did.
"Royal names are diminishing. Edward VIII had 7; Prince Charles has 4; Prince William 4; Prince George 3. #princegeorge," he tweeted.
Charles Mosley, author of Blood Royal, told the BBC he knew it would be George.
"It was the name of the sovereigns under whom this country became a world player - in the 19th Century, defeating Napoleon, and in the 20th Century, winning two world wars.
"Louis is a nod in the direction of the Mountbatten element in the Royal Family. And indeed the surname of the royal family is still Mountbatten. His (Prince George's) birth certificate and marriage certificate will say Mountbatten-Windsor."
As well as being an established regal name, George is also a saint's name. St George was an early Christian martyr and is the patron saint of England.
The world finally knows how to address the latest addition to the British royal family: HRH Prince George Alexander Louis of Cambridge.
The baby boy is third in line to the throne, after grandfather Prince Charles and father Prince William. Here's some of the history behind the young prince's three names:
Royal Baby: What's next?
George
"George" -- the front-runner before the announcement, according to many UK bookmakers -- was the name of Queen Elizabeth's father, King George VI, who reigned from 1936 until his death in 1952. He assumed the throne on the abdication of his brother, Edward VIII. His life was depicted in the Oscar-winning movie "The King's Speech."
Baby's name is steeped in royal history The royal aunt and uncle Excited crowds welcome royal baby boy
Photos: Other famous Georges Photos: Other famous Georges
George I, born in Germany, became king in 1714. He was followed by a line of kings with the same name, including George III, who was known for his bouts of insanity.
The name is also a patriotic choice for many in the UK: Saint George, patron saint of England, is known for his legendary defeat of a dragon in the third century. His feast day is celebrated on April 23, (the date also associated with the birth of William Shakespeare, England's most revered writer).
Despite its royal connections, George has humble origins, derived as it is from the Greek name "georgios" meaning "earth worker" or "farmer."
Other historical Georges: composer George Frideric Handel (1685-1759), first president of the United States George Washington (1732-1799), and the Pacific explorer George Vancouver (1757-1798). Authors Mary Anne Evans and Eric Arthur Blair also chose George as their pen names: George Eliot (1819-1880) and George Orwell (1903-1950) respectively.
Alexander
This gallant title means "defender of men," from the Latin form of the Greek name "Alexandros." Alexander III of Macedon (356-323 B.C.), better known as Alexander the Great, courageously ruled and conquered many parts of the world before his untimely death at age 32.
The name "Alexander" is a feature of the Dutch royal family: Willem-Alexander ascended to the throne after the abdication of his mother Queen Beatrix earlier this year.
Other historical Alexanders: English poet Alexander Pope (1688-1744), American statesman Alexander Hamilton (1755-1804), Scottish-Canadian explorer Alexander MacKenzie (1764-1820), Russian poet Alexander Pushkin (1799-1837) and Alexander Graham Bell (1847-1922), the Scottish-born inventor of the telephone.
Louis
The name "Louis" originates from the English and French interpretations of the German name Ludwig, which can be interpreted as "renowned warrior."
Louis was the first name of Lord Mountbatten, uncle of George's great-grandfather Prince Philip, Duke of Edinburgh, and mentor to Prince Charles. He was killed by the IRA while holidaying in Ireland in 1979.
No British monarch has been named Louis, but it is very popular across the English Channel in France, where 18 kings have taken the name from 814 onward. Louis XIV, the Sun King, reigned from 1643 until 1715 and was hailed by many as the greatest monarch of his age because of the growth in French power and the opulence of his court, which included the Palace of Versailles.
Louis XVI was the king of France from 1774 until 1792, when he was found guilty of treason after the revolution and executed in 1793.
Other historical figures named Louis: French scientist Louis Pasteur (1822-1895); Métis leader Louis Riel (1844-1885), who led a rebellion against Canada; and Scottish author Robert Louis Stevenson (1850-1894), who wrote "Treasure Island" and "Dr. Jekyll and Mr. Hyde."