Rub a lime wedge on rims of 8 glasses, then dip rims in superfine sugar.
Cut 3 melon quarters into 1-inch chunks. Puree melon in a food processor until smooth. Strain puree through a cheesecloth-lined sieve, and discard solids. (You should have about 4 cups.) Using a 1-inch melon baller, scoop balls from remaining quarter, and reserve.
Bring granulated sugar and water to a boil in a small saucepan, stirring until sugar dissolves. Remove from heat, and add mint. Let stand at room temperature, uncovered, for 10 minutes. Strain through a fine sieve, and discard mint leaves. (You should have a scant 1/3 cup syrup.)
Combine melon puree, mint syrup, rum, and lime juice in a pitcher, and refrigerate until cold, about 30 minutes. Add ice to pitcher, and divide among glasses. Garnish each with melon balls, mint sprigs, and lime wedges, and serve immediately.