Combine the sugar and 2 cups water in a small saucepan and bring to a simmer over medium heat. Cook, stirring, until the sugar dissolves, about 4 minutes; transfer to a blender and let cool completely.
Add the mint to the blender and puree until smooth. Strain through a fine-mesh sieve into an 8-inch-square stainless-steel or glass baking dish. Finely grate the skin of the apple into the pan (you won't need the rest of the apple), then stir in the lime juice.
Freeze until ice crystals begin forming around the edges, about 45 minutes. Use a fork to scrape the crystals toward the center of the pan, then continue freezing, scraping every 30 minutes, until frozen, about 4 hours.
Avoid aluminum pans-they can react with the acidic ingredients and produce a metallic taste. Use stainless steel or glass instead.