Combine cranberries, 1/2 cup water, and 1/2 cup sugar in a medium saucepan; bring to a boil. Cook over medium heat 8 minutes, stirring frequently. Remove from heat, and cool slightly. Process cranberry mixture through a food mill; discard pulp, and reserve puree.
Combine cranberry puree, remaining water, sugar, and juice; stir until well blended. Pour cranberry mixture into two 2-cup metal ice trays (with dividers removed); freeze until ice crystals form around upper edge of ice trays. Remove from freezer; stir well, and return to freezer. Repeat procedure until mixture reaches the consistency of a sherbet. Scoop into sherbet dishes, and serve immediately.