In the 2006 video game The Godfather (based on the 1972 film), the game ends immediately after you become Don of New York City.
The Godfather II (2009) features a strategic "Don's View" mode allowing ongoing empire management after the main campaign, but the first game is strictly linear to the conclusion.
Tip: Complete all side activities (assassinations, favors, racing) before triggering the final mission if you want 100% completion.