Place berries in the work bowl of a food processor and puree until smooth. Combine berry puree, sugar, and salt in a large saucepan. Heat until melted and just simmering. Whisk cornstarch into the cold water; stir into heated berry mixture. Remove from heat, and stir in lemon juice. Cool slightly. Refrigerate berry mixture until cold, about 2 hours.
Freeze in ice cream maker according to manufacturer's instructions.