Combine the water and sugar in a large saucepan, and bring to a boil, stirring until the sugar dissolves. Remove from heat, and stir in honey. Cool completely.
Grate ginger; place ginger on several layers of damp cheesecloth. Gather edges of cheesecloth together; squeeze cheesecloth bag over a small bowl to measure 1 tablespoon ginger juice. Combine the sugar mixture, ginger juice, and lemon juice in a large bowl.
Pour mixture into the freezer can of an ice-cream freezer, and freeze according to manufacturer's instructions. Spoon the sorbetto into a freezer-safe container; cover and freeze 1 hour or until firm. Garnish with lemon slices, if desired.