Stir sugar and 1 cup water in small saucepan. Bring to boil over high heat, stirring until sugar dissolves. Boil 1 minute.
Transfer syrup to large bowl. Chill until syrup is cold, about 3 hours.
Working in batches, puree blackberries with juices and cold syrup in blender until smooth. Strain into another large bowl; discard seeds. Stir in lemon juice.
Process berry mixture in ice cream maker according to manufacturer's instructions.
Transfer sorbet to container; cover and freeze until firm, about 6 hours. (Can be made 1 week ahead. Keep frozen.)
Using large spoon, scoop out flesh from watermelon, leaving rind intact and forming bowl (reserve melon for another use or serve alongside sorbet, if desired).
Drain excess juice from watermelon bowl. Cover and chill watermelon bowl until cold.
