Mendelsohn left the agency to join Grey Worldwide London in March this year.
Exon, a business director on the Barclays and Warburtons accounts, joined BBH in June 1994 from Ogilvy & Mather, where he started his advertising career. He worked on the Audi account for six years, during which time he wrote an award-winning IPA paper on the client.
He was appointed to the board in 1999 and recently ran the successful pitch for the pan-European Sandford pens account.
Exon will report to the UK managing director, Gwyn Jones, who said: "New business is one of those funny areas where you can't keep transferring into new roles because it undermines the credibility of the seller.
"So it's important to us that our head of new business is someone who has real experience and knowledge of the agency. Richard is part of the culture here, and as such he is perfect for the role."
The global new-business role will continue to be handled by Hugh Baillie.
Exon said: "BBH's greatest strength has always been creative selling - generating original brand and advertising ideas that are truly effective. I'm delighted to have the opportunity to spread this message as widely as possible and encourage new clients to join BBH."