Harris replaces Elliot Moss, who left the agency a month ago to join one of its clients, the law firm Mishcon De Reya, as the head of a joint venture between the two businesses.
In his new role, Harris will take responsibility for overseeing the management of the agency's client base, as well as strengthening its new-business performance.
Harris first joined BBH in 1990 as head of account management after cutting his teeth at Saatchi & Saatchi.
After a short spell away from the agency, at SP Lintas London and the Lowe Group, Harris took on the role of group business director at BBH London in 1996 before moving across to BBH Asia- Pacific to take the position of chief executive two years later.
In his most recent role as president of international operations for BBH, Harris led international client links with brands such as Diageo, British Airways and LG Electronics.
Margaret Johnson, the group chief executive at Leagas Delaney, said: "He is a down-to-earth character who shares our values of creativity and strategic thinking."
Harris added: "To lead and help shape this into a future-facing business is a very exciting opportunity."