Place berries in the bowl of a food processor and purée until smooth. Pour mixture into a large saucepan over medium heat. Add sugar and salt; stir until dissolved and bring to a simmer.
Whisk together cornstarch and cold water in a small bowl; stir into simmering berry mixture. Remove the saucepan from heat, stir in lemon juice, and allow to cool slightly. Refrigerate berry mixture until cold, about 2 hours.
Freeze in an ice cream maker according to manufacturer's instructions.