Kate, 31, arrived at the private Lindo Wing at St. Mary's Hospital in Paddington, London just before 6am on Monday morning (22 July).
She was accompanied by husband Prince William.
David Cameron has said: "Best wishes to them, a very exciting occasion and the whole country is excited with them, so everyone is hoping for the best."
Ever since Kate and William announced the pregnancy in December 2012 the world has been gearing up for the birth of the royal baby; the third-in-line to the throne.
"Royal baby", "Kate Middleton", "Kensington Palace" and "Duchess of Cambridge" have all been trending topics on Twitter this morning, with celebrities and royal fans tweeting their excitement all morning.
Buckingham Palace have not commented further than confirming Kate is in labour.
A statement this morning said: "Her Royal Highness The Duchess of Cambridge has been admitted to St. Mary's Hospital, Paddington, London in the early stages of labour. The Duchess travelled by car from Kensington Palace to the Lindo Wing at St. Mary's Hospital with The Duke of Cambridge."
The palace later confirmed the labour is "progressing as normal".
The world will learn of the royal baby's arrival when the official birth notice is taken from the hospital to Buckingham Palace, where it will be displayed on an easel for passersby to see.
Kate Middleton's pregnancy in pictures: