Set a large fine-mesh sieve over a large pitcher or jug. In batches, in a food processor or blender, puree melon until smooth. Pour melon puree through sieve, pressing on solids with a rubber spatula (you should have about 4 cups juice).
In a small bowl, combine superfine sugar and fresh lime juice; stir until sugar dissolves. Add lime mixture and 4 cups water to melon juice and mix well. Adjust sweetness with more sugar if desired. (Mixture can be refrigerated, up to 2 days.) To serve, add plenty of ice and 2 cups raspberries.