In a heavy medium saucepan, combine sugar, the water and corn syrup. Cook and stir over medium-high heat about 5 minutes or until sugar is dissolved and mixture comes to boiling. Remove from heat; stir in lime juice. Cool slightly. Transfer syrup to a 2-cup glass measuring cup; set aside.
Place half of the melon cubes in a food processor or blender. Cover; process or blend for 30 to 45 seconds. With processor or blender running, add half of the warm syrup through the processor's feed tube or the hole in the blender's lid; process or blend until almost smooth. Transfer the fruit puree to a 3-quart rectangular baking dish. Repeat processing or blending the remaining melon and syrup.
For refrigerator-freezer sorbet: Cover the baking dish and place in the freezer, making sure dish is level. Freeze for 5 to 6 hours or until almost firm, stirring once. Using a fork, break up the frozen mixture into chunks. Transfer chunks to a large, chilled mixing bowl. Beat with an electric mixer on medium speed until smooth and fluffy but not melted. (Or place in a food processor or blender, half at a time, if necessary. Cover and process or blend for 30 to 60 seconds or until smooth and lightened in color.) Return quickly to the cold baking dish, spreading evenly. Cover and freeze for 6 to 8 hours more or until sorbet is firm.
For ice cream freezer sorbet: Cover the baking dish and chill in the refrigerator for 3 or 4 hours or until chilled. Pour fruit puree mixture into a 1 1/2- or 2-quart ice cream freezer. Freeze according to the manufacturer's instructions. Transfer sorbet to a freezer-safe, airtight container; let ripen in the freezer for 3 to 4 hours or until sorbet is firm.
To serve, let stand for 5 to 10 minutes at room temperature before scooping.