Marr has been political editor at the BBC since May 2000. He also presents 'Start the Week' on Radio 4 and other current affairs TV and radio including BBC One's 'Panorama' political strands.
The new as-yet-to-be-named series with Marr at the helm, will broadcast in the same slot as 'Breakfast with Frost' from September.
Helen Boaden, director of BBC News, said of Marr: "He has made a huge impact on our audiences who appreciate his wit and vitality as well as his tremendous knowledge, analysis and insight."
'Breakfast with Frost' is coming to the end of its 12-and-a-half year run at the end of this month -- also marking the 500th show presented by Frost.
A replacement Sunday morning series has been lined up for the summer featuring BBC News 24 presenter Peter Sissons. The show is to be called 'News 24 Sunday'
With Marr's departure, the coveted role of BBC political editor is being advertised internally at the corporation and externally.
Possible candidates include BBC economics editor Evan Davis and BBC political correspondent Laura Trevelyan.
Frost's departure comes as other veterans in the industry also take a bow. Channel 4 News' political editor Elinor Goodman is to leave the programme after more than 23 years with the show.
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