The managing director role has been vacant since Paul Longhurst departed in 2001. Since then, the commercial director, Karen Mayer, has been overseeing the running of the agency.
Clays joined BLM in 2000 as the director of interactive from PHD, where he was a TV group head. Before that, he worked in the TV department at Carat.
He will lead a team of 20 staff, who work across clients including Wanadoo, Thomas Cook, Intelligent Finance, Reebok and Domino's Pizza.
The agency recently added the online accounts for Suzuki and Nikon to this list.
Steve Booth, the BLM chief executive, said: "Following a succession of new-business wins and continued heavy investment by our clients into digital media, Quantum has moved on to another level, which Dan has been at the heart of for some time."
Quantum has also appointed the former Carat account director Dom Finney to an account director role. He will be charged with co-ordinating the agency's buying effort and developing its research and analysis division, Quantum Intelligence.
Aiden Mark is also joining the agency from Starcom Motive as a senior planner.
Clays said: "Everyone here has a very clear perspective on the service we are able to offer, which is all about making a sizeable impact on clients' business through online media. Advertisers need more choice in this area than just the default solution of their existing offline agency."
Following Clays' promotion, Mayer will return to concentrating on the agency's commercial operations.