His girlfriend Cath Mason, 43, gave birth to a baby boy earlier this week. Frank, 55, announced the happy news live on his Absolute Radio show, revealing the couple have named their new arrival Buzz Cody.
He weighed in at 9lb exactly.
Explaining the couple's unusual name choice, Frank said: "Now I know celebrity types get condemned for giving their babies unusual names.
"So this is the name that I went for. He's called Buzz B-U-Z-Z. If you're my age, obviously he's after Buzz Aldrin, the second man on the moon. For you youngsters listening, yes, it's that puppet from Toy Story.
"His second name is Cody, as in Buffalo Bill Cody, the greatest showman of all time, so I've worked out his career."
He also revealed Buzz was born after a 5am dash across London to hospital, admitting: "Cath was already in mighty labour when we arrived. We had the exciting five o'clock drive across London with me very worried about her and slightly worried about the upholstery!"