Starting in October, Cobbold will be charged with reviving the agency and will report to Kevin Park, the chief executive of Cheil Europe.
The chief executive post at BMB has been vacant since the end of last year when Juliet Haygarth stepped down to recover from health issues.
Cobbold was previously the managing director at US-based innovation agency Redscout, where he launched the London office in 2014 and worked with clients including Boots, PepsiCo, Warner Brothers, Marriott, Pernod Ricard and Hello Fresh.
Prior to Redscout, Cobbold was the deputy managing director at Publicis London and global business director at Bartle Bogle Hegarty on the Johnnie Walker account.
Cobbold said: "Few agencies are held in the same regard and affection as BMB, and I will be building on this with Matt and the team. Together, we’ll be driving a business that serves the needs of today’s marketers and brand owners."