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.