Combine the sugar and water in a saucepan and heat over medium heat. As soon as the sugar dissolves, turn the heat off, add a handful of fresh mint sprigs, and allow the syrup to cool to room temperature.
When the syrup cools to room temperature remove the mint, add the lime juice, and then pour the mixture into a wide, shallow, freezer-safe dish. Put the dish in the freezer. After 45 minutes, take the dish out of the freezer and stir it with a fork. Return the dish to the freezer.
After 30 minutes, take the dish out of the freezer and stir with the fork again before returning the dish to the freezer.
Continue freezing the granita, stirring it with a fork every 30 minutes, until completely frozen, about 3 to 4 hours total.
Serve with fresh mint leaves, if desired.