1. Combine first 3 ingredients in a saucepan over medium-high heat; bring to a boil, stirring until sugar dissolves. Remove from heat; add lemon juice and mint. Cover and chill.
2. Strain juice mixture through a sieve into a bowl; discard solids. Pour mixture into the freezer can of an ice-cream freezer; freeze according to manufacturer's instructions. Spoon sorbet into a freezer-safe container; cover and freeze 1 hour or until firm. Serve with blackberries; garnish with lemon slices, if desired.