Conti, a previous Olivier and Tony award winner, will begin performances on 31 March, taking over from Martin Shaw.
Twelve Angry Men, which opened in the West End in November, has extended its run three times, most recently until 14 June 2014. The cast also features Robert Vaughn and Jeff Fahey, in a production directed by Christopher Haydon.
Originally written, by Reginald Rose, as a television play in 1954, Twelve Angry Men was adapted for the stage in 1955 and as an Oscar-nominated Hollywood film, produced by and starring Henry Fonda in 1957.
It sees 12 jurors deciding the fate of a young delinquent accused of killing his father. But what appears to be an open and shut case soon becomes a dilemma for the 12, as their prejudices and preconceived ideas about the accused, the trial and each other turn the tables every which way.
Tom Conti is best known for starring roles in films such as Merry Christmas Mr Lawrence and Shirley Valentine, while recent credits include The Dark Knight Rises and Street Dance. He won an Olivier Award and a Tony Award for Whose Life is it Anyway? and a Best Actor Oscar nomination for his role in Reuben Reuben.