1 cup granulated sugar
2 cups packed fresh mint leaves
2 1/2 cups fresh lime juice (about 20 limes), plus 4 limes, thinly sliced crosswise, for garnish
1 1/2 cups superfine sugar
Bring granulated sugar and 1 cup water to a boil in a medium pan over high heat. Stir until sugar is dissolved. Refrigerate until chilled, about 1 hour. Add sugar syrup and mint to a blender; puree. Pour through a very fine mesh sieve into a large serving container; discard solids.
Add lime juice, superfine sugar, and 2 quarts water to a serving container. Stir until sugar is dissolved. Refrigerate until ready to serve, up to 1 day.
To serve, add ice cubes to fill and garnish with lime slices.