The Location, Location, Location presenter had initially challenged editor Paul Dacre to a debate on journalism after a feature on the Daily Mail website said that she "appeared to be sporting a moustache".
Allsopp told Radio 5 Live that she in fact suffered from a hormonal condition not uncommon in new mothers that causes a discolouration of the skin.
She later told the BBC: "I'm not interested in sticking it to the Mail. All I'd say is, you got it wrong this time - back up and pull your socks up."
On her Twitter page, she added: "Proper apology from the Mail, piece off site. As I say to the kids and must stick to myself, once you've said sorry we'll say no more."
A Daily Mail spokesperson confirmed that it had spoken to Allsopp and that the matter had been "resolved".