The first – and only – remake to win Best Picture is The Departed (USA, 2006), Martin Scorsese's (USA) remake of Infernal Affairs (Hong Kong, 2002).
The Coen brothers' (USA) remake of True Grit (USA, 2010) was nominated for the 2010 Best Picture Oscar but lost out to The King’s Speech (UK, 2010). A remake is defined as a new version of an existing movie, not simply a movie made using the same source material as a previous film. E.g., Mutiny on the Bounty (USA, 1963) won Best Picture but is not a remake of the earlier film The Mutiny of the Bounty (Australia, 1916) - it is an original movie inspired by the same historical event.
Comments below may relate to previous holders of this record.