Bring sugar and 1 cup water to boil in a saucepan. Turn off heat, add mint, and cover. Let stand 5 minutes. Pour through a fine wire-mesh strainer into shallow, plastic, 1-gallon container; discard mint. Stir in grapefruit juice.
Freeze several hours or overnight. Let stand 10 to 15 minutes before serving. Scrape and shave granita with a fork or ice cream scoop. Serve immediately.