Rich, 60, will step down from his post as the vice-chairman of MediaCom Europe. In 1998, he was responsible for the merger of The Media Business with Grey's MediaCom, and became the consolidated agency's chief executive.
Having founded The Media Business in 1975, Rich built it into a sizeable agency before the merger. He moved to the wider European role in 2000.
Rich said he had no intention of retiring, and would instead continue to work in advertising, though in an area that would be non-competitive with MediaCom.
MediaCom has not yet decided on whether to appoint a successor to Rich in the European role.
Rich said: "I look at the team (at MediaCom) and think they're brilliant and that the future is safe and sound."
Alexander Schmidt-Vogel, MediaCom's worldwide chief executive, added: "In the process of The Media Business merger, Allan was key. Most mergers fail but in this case you could say it was perfect. Allan was key to the success in bringing people together and creating a new culture because he's excellent with people."