Combine water, Xylitol and preserves in medium mixing bowl, stirring until jam dissolves.
Puree blackberries (or raspberries) with lemon juice in blender or food processor. To remove seeds, force through a fine strainer into bowl. Add sweetener mixture and stir well to combine. Chill until cold.
Pour into shallow metal cake pan and put in freezer until solid; about 6 hours.
Break into chunks and process in a food processor until smooth