Cooper will make his directorial debut on the film, which will also mark Lady Gaga's first lead role in a feature.
Lady Gaga will compose and perform new songs in the film, which tells of an aspiring starlet's relationship with a leading man whose star is on the wane.
Previous versions of the story starred Judy Garland and Barbra Streisand.
Lady Gaga, who was nominated for an Oscar for song 'Til It Happens To You, has previously starred in films including Machete Kills and Sin City: A Dame to Kill For.
Earlier this year, the 30-year-old won a Golden Globe for her work in the latest season of US anthology series American Horror Story.
Writing on Twitter, Gaga - real name Stefani Germanotta - said she was "elated" to be working with "brilliant visionary artist" Cooper.
In a statement, Warner Bros's Greg Silverman said "the world [was] in for a treat as these great artists craft an all new vision of A Star is Born."
Clint Eastwood had originally been slated to direct the film, but that version hit the buffers when Beyonce withdrew from the project.
Cooper, 41, received Oscar nominations for his performances in Silver Linings Playbook, American Hustle and American Sniper, which he also produced.
According to Deadline, production on A Star is Born will begin early next year in California.
Garland appeared with James Mason in the 1954 iteration of A Star is Born, itself a remake of a 1937 film starring Janet Gaynor and Fredric March.
Streisand starred alongside Kris Kristofferson in the 1976 version, which transplanted the story from Hollywood to a music industry setting.