Place the cubed melon in a shallow container and freeze.
Place the 1 cup of chopped mint, sugar, and water in a small saucepan over medium heat. Stir until the sugar is dissolved. Bring to a boil and simmer undisturbed, for 2 minutes. Pour syrup through a fine-mesh sieve, pressing hard on the solids, and refrigerate until chilled.
Place the frozen melon in the bowl of a food processor. Add remaining 1/4 cup of mint leaves, the cold mint syrup, and process until smooth. Serve immediately, or store in an airtight container for up to 2 weeks.