In a saucepan over medium heat, add the sugar, water and mint leaves. Cook for about 5 minutes, stirring often, until the sugar is dissolved and the syrup is clear. Set aside to cool.
Put the sugar syrup, lime juice, mint leaves, lime zest, and rum into a blender and blend until smooth. Add the ice and blend until slushy. Spoon into glasses.
Garnish with a sprig of mint and a lime wedge if desired.
