In a large saucepan, bring water and sugar to a boil. Cook and stir until sugar is dissolved. Cool slightly. In batches, process the sugar syrup and watermelon in a food processor. Transfer to a large bowl; stir in lemonade concentrate. Cover and refrigerate until chilled. Fill cylinder of the ice cream freezer two thirds full' freeze according to manufacturer's directions. Refrigerate remaining mixture until ready to freeze. Allow to ripe in ice cream freezer or firm up in the refrigerator freezer for 2-4 hours before serving.