He will replace Steve Blamer, who is returning to the US to run Grey's New York operation, and will work in close partnership with Tim Mellors, Grey's executive creative director.
The news comes only three weeks after the announcement that Blamer is leaving London a year early. Smith will join Grey at the end of next month and will have a brief handover period before Blamer's departure.
"If I had failed to find the right person, I'd have done the job myself," Mellors, who was key to the hiring, said. "We have been bedding down the agency and sorting out its new-business performance, and now that we are on top of that we need to get to grips with the creative product."
Mellors added: "When I came here I said it would take three years to turn the creative product around. Martin's priority will be working to put creative first."
Smith's resignation comes almost exactly 18 years after he joined BBH in 1982 as a board director. He was the new-business director from 1987 to 1988 and became vice-chairman in July 1988.
In January 1989 Smith became chairman of BBH's below-the-line group of companies. In July 1996 he was appointed managing director before taking his current role in June 1998.
Smith began his career in advertising as a copywriter at Ogilvy Benson & Mather and he made the shift to account management in 1976.
"Many agencies say that they want to improve but they are just paying lip service to the idea," Smith said. "Tim means business and what he has achieved with Steve Blamer so far proves that."
He added: "I've been at BBH for 18 years and after a while you feel that you want to go and run something on your own."
Blamer and Ed Meyer, Grey's worldwide president, are due to make an announcement on this and other changes later this week.