Combine the water and the sugar in a heavy medium saucepan over medium-high heat. Stir until the sugar is dissolved. Then bring to a boil and boil for 5 minutes without stirring. Remove from the heat.
Place the torn mint leaves in a medium mixing bowl and pour the hot sugar mixture over them. Let the mixture steep for 45 minutes. Then strain, pressing down hard on the mint leaves to extract as much liquid as possible, into a large nonreactive mixing bowl. Discard the mint leaves. Stir in 4 teaspoons lime juice.
Add the cubed cantaloupe and honeydew to the bowl and toss to coat. Cover bowl and refrigerate for at least 3 hours or overnight. Strain 11/2cups cubed cantaloupe and honeydew melon from the mixture for the garnish. Purée the soup in a food processor, blender or food mill, and return it to the bowl. Or, use an immersion blender to purée the soup in the bowl. Taste the soup and if you want a more pronounced citrus flavor, add 1 to 2 teaspoons additional lime juice. Cover the soup and the reserved melon cubes, and refrigerate until ready to use. (Soup can be prepared 1 day ahead.)
Thirty minutes before you plan to serve the soup, place 4 soup bowls in the refrigerator to chill. To serve, divide soup evenly and ladle into the bowls. As a garnish, mound some of the reserved cubes of cantaloupe and honey dew melon into the center of each serving and tuck in a mint sprig.