In a saucepan over medium heat, add the sugar and water. 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 slice.
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