A Bronx Tale, adapted from the film of the same name, is set against the backdrop of organised crime and racial tension in 1960s New York.
The world premiere will take place at the Paper Mill Playhouse in New Jersey on 4 February.
It will be co-directed by Jerry Zaks, the man behind Broadway hits such as Guys and Dolls, with music by multiple Oscar-winner Alan Menken.
A Bronx Tale began life as a one-man stage play, starring Chazz Palminteri as an Italian-American teenager who must choose between his father and a mob-boss father figure.
An aspiring actor, he had become frustrated with his inability to win a leading role and decided, "If they're not gonna give me a great part, I'll write one myself."
Palminteri played eighteen parts, winning widespread acclaim in Los Angeles. Shortly before the play transferred to New York, De Niro bought the film rights.
He cast Francis Capra in the lead role, with Palminteri as the mafia figure. De Niro, who played the teenager's father, also directed.
Plans to turn the show into a musical with the help of Billy Joel never came to fruition, and Palminteri - who had gone on to appear in The Usual Suspects and Bullets over Broadway - revived his one-man show on Broadway in 2007.
That led to renewed interest in the musical version, with Alan Menken eventually signed up to write the score.
The composer previously wrote songs for Disney films including The Little Mermaid, Beauty and the Beast, Aladdin, and Pocahontas.
Paper Mill's new season will launch with big band musical Bandstand in October.
A Christmas Story, based on the 1983 movie of the same name, will run throughout the holiday period at the end of the year.
The season will also feature Pump Boys and Dinettes and West Side Story.
"We are proud to have created an irresistible line-up of stories you will love, glorious music that will warm your heart and many memorable moments of theatre magic," said Paper Mill's producing artistic director Mark S Hoebee in a statement.
Actor Robert De Niro is to direct a new stage musical in New Jersey next year.