Stir sugar and water in small saucepan. Bring to boil over high heat, stirring until sugar dissolves. Boil 1 minute. Transfer syrup to large bowl. Chill until cold, about 3 hours.
Puree blackberries with juices and cold syrup to taste in blender until smooth. Strain through fine-meshed sieve into another large bowl; discard seeds. Stir in lemon juice or Limoncello. (The addition of a little alcohol keeps the sorbet's texture scoopable in the freezer.).
Process berry mixture in ice cream maker according to manufacturer's instructions. Transfer sorbet to container; cover and freeze until firm, about 6 hours.