The decision ends weeks of speculation about who was to replace Allan, who is moving across to head WPP's Group M with responsibility for MediaCom, MindShare and Mediaedge:cia.
Lawson was marketing director at MediaCom before being promoted to joint managing director with client services director Jane Ratcliffe in 1999.
Other internal candidates in the running included Ratcliffe and MediaCom Direct managing director David Kyffin.
Ratcliffe has reportedly been promoted to sole managing director of MediaCom in the UK and Kyffin is tipped to follow Allan to Group M.
At the same time, it is expected that Kelly Clark, MindShare's UK chief executive, will take up the role of Group M European chief executive.
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