Charles said the arrival of the Duke and Duchess of Cambridge’s baby son, their third child, was a “great joy”.
The heir to the throne, who turns 70 this year, is also grandfather to Prince George and Princess Charlotte.
Charles said: “We are both so pleased at the news.
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“It is a great joy to have another grandchild, the only trouble is I don’t know how I am going to keep up with them.”
William and Kate are settling into life as a family of five after bringing their new baby son home from hospital.
They have retreated to the sanctuary of Kensington Palace after introducing their third child to the world on the steps of the exclusive maternity unit, the Lindo Wing.
The name of the baby prince of Cambridge, who was born on Monday at 11.01am weighing 8lb 7oz, has yet to be unveiled.
Royals traditionally keep the public guessing before revealing their choice.
William and Kate will want to share the name with the Queen and the rest of their family before making it public.
Kensington Palace said the name would be announced in “due course” but the favourite at the bookies is currently Arthur, followed by James, Philip and Albert.
The new baby prince’s great-grandmother the Queen, 92, is still at Windsor Castle for Easter Court, with the Duke of Edinburgh who is recuperating following a hip operation.
Born fifth in line, the baby is the Queen and Philip’s sixth great-grandchild.
William joked on Monday how he had “thrice the worry” now as he took the prince home to Kensington Palace.
Quizzed on whether they had decided on a name, the duke added: “You’ll find out soon enough.”