Pour grapefruit juice into a 9-by 12-inch nonreactive glass baking pan. Place in the freezer. Freeze 6 to 7 hours, or until almost solid but still somewhat slushy. While juice is freezing, rake with a fork every hour or so to develop large, snow cone-like crystals.
To serve, use an ice cream scoop, and scoop into glasses. Pour 1 ounce of vodka over each ball. Garnish with mint and serve.