"The Duke and Duchess of Cambridge are pleased to announce that they named their son Louis Arthur Charles".
The second son of the Duke and Duchess of Cambridge, who are known as the Earl and Countess of Strathearn in Scotland, he will be known as His Royal Highness Prince Louis of Cambridge.
Experts correctly predicted Louis would be among the names in honour of the Prince of Wales's late great-uncle Lord Mountbatten, as well as Philip in a nod to the Duke of Edinburgh.
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The name pays tribute to William's great-great uncle Louis Mountbatten, who was killed by the IRA in 1979, while Charles is the name of William's father.
He is the fifth in line to the throne, after his grandfather Prince Charles; his father; and his siblings, Prince George and Princess Charlotte.
The couple stepped out of St Mary's Hospital in central London to cheers from a crowd of supporters and global media outlets gathered outside.
Although the name had not been officially announced, the Rector of Sandringham made a mention in a prayer for "Peter and Autumn Phillips and their daughter Savannah".
Alexander is also one of Prince George's middle names, as his full name is George Alexander Louis.More news: After 35-year hiatus, ABBA announces new music
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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.
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The name does not have a strong British royal pedigree, being more commonly associated with kings of France.
Prince William was there for the birth, as he was for their previous two children.
Australian High Commissioner Alexander Downer also suggested Alexander, to which William said cryptically: 'Funny you should say that.It's a good name, ' The Sun reports.
Diana was thought to have preferred names such as Sebastian and Oliver, while Charles was reportedly holding out for Albert.
Out of courtesy, the Queen has to be told what has been decided before a name is announced.
Prince Louis has been with his parents and relatives at Kensington Palace since Monday evening.