In a large saucepan, mix the sugar, cocoa powder, vanilla, cinnamon, and salt. Stir in 2 cups water and the espresso. Cook over low heat until the ingredients are dissolved. Off the heat, stir in the coffee liqueur. Transfer to plastic containers and refrigerate until very cold.
Freeze the mixture in an ice cream freezer according to the manufacturer's directions. The sorbet will still be soft; place it in a plastic container and freeze for 1 hour or overnight, until firm enough to scoop.
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