Kate Middleton's parents, Carole and Michael Middleton, were the first visitors to see the new baby prince.
The yet to be named first baby of the Duchess of Cambridge and Prince William was born on Monday afternoon (Tuesday morning NZT) in London.
Asked if she would reveal the name or had made any suggestions, Kate's mother said: "Absolutely not."
The baby is third in line to the throne after grandfather Charles and William, 31, and pushes the fun-loving Prince Harry, William's brother, into fourth place.
"I'm thrilled," Charles said. His wife, Camilla, spoke of a "wonderfully uplifting moment for the country", saying Charles would make a "brilliant" grandfather.