Combine the sugar, water, and ginger in a saucepan over medium heat. Gently simmer and stir for 2 minutes until the sugar is dissolved ¿ do not allow the syrup to boil or get dark. Remove from the heat, cover, put in the refrigerator, and let steep until cold.
Strain the lime juice into a blender. When the syrup is cold, fish out the ginger slices and pour the syrup into the blender. Add the lime zest. Then add the chopped mint and crushed ice and process until slushy. Spoon the slushy into bowls, garnish with mint leaves, and serve immediately.