Make the Black Raspberry syrup - Add the water and sugar to a medium saucepan over medium heat, and stir until the sugar has dissolved. Stir in the black raspberries. Let the syrup cool a bit, then process in a food process for a few seconds to break down the berries.
Strain the syrup and chill - Strain through a fine mesh strainer to remove the seeds. Stir in the lemon juice and the vodka, then stash the mixture into the fridge until it's completely cooled.
Churn and freeze - Once the mixture is cool, churn it in your ice cream maker, following the manufacturer's instructions. Move the the freezer and freeze for at least two hours before serving.