The child is the first person to hold the title prince of Cambridge. Kate and William are known as the duchess and duke of Cambridge.
A statement released from Kensington Palace said the couple went to the hospital by car without a police escort.
The duchess, the former Kate Middleton, was being attended to by a medical team led by Queen Elizabeth II's former gynecologist, Marcus Setchell, and assisted by Alan Farthing, the queen's current gynecologist.
William and Kate married in April 2011.
Kate Middleton recycles dress at movie premiere
Beyonce flaunts bikini body, Blue Ivy in vacation pics