Puree blackberries, aspartame sweetener, and lemon juice in a food processor or blender.
Measure 1 1/4 cup (310 mL) of the blackberry puree and place in a heavy saucepan.
Reserve remaining blackberry puree for the sauce.
Cook and stir over medium heat until mixture is reduced to a scant 1 cup (250 mL).
Transfer to a bowl and chill for at least 30 minutes.
Line a 9 x 5 in. (23 x 12.5 cm) loaf pan with plastic wrap.
Transfer 1 1/2 cup (375 mL) of the lemon sherbet to a large bowl.
Fold in the reduced blackberry puree.
Spread one-third of the remaining lemon sherbet in the bottom of the prepared loaf pan.
Cover with the blackberry-lemon mixture.
Top with the remaining lemon sherbet.
Smooth the top and freeze overnight.