Combine fruit juice and sugar in saucepan. Bring to boiling and boil 1 minute.
Remove from heat and cool. Put berries in blender, add 1/4 cup of the cooled juice mixture and process until smooth.
Combine puréed berries, remaining juice mixture and lime juice; stir well. Pour mixture into freezer container of a hand-turned or electric ice-cream maker. Freeze according to manufacturer's directions.
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