The 53-year-old political commentator was taken to hospital yesterday (January 8), and is currently responding to treatment.
"Andrew Marr was taken ill yesterday and taken to hospital," a statement read. "The hospital confirmed he has had a stroke.
"His doctors say he is responding to treatment. His family have asked for their privacy to be respected as he recovers."
The statement added that his TV and radio shows will be hosted by guest presenters while he recovers.
"We will continue to broadcast The Andrew Marr Show and Radio 4's Start The Week with guest presenters in his absence.
"His colleagues and the whole BBC wish him a speedy recovery."
Marr has been hosting his Sunday morning television programme since September 2005.
He was the editor of The Independent from 1996-1998, and was political editor of BBC News from 2000-2005.