Cook the syrup. In a small sauce pan combine the blackberries, sugar, and water over medium-high heat. Bring to a boil, then reduce the heat to low and simmer for 2 minutes.

Blend the syrup. Remove the syrup from the heat. Transfer to a blender or use a handheld immersion blender to puree the berries. Strain into a container.

Make the cocktail.  In a cocktail shaker, combine the vodka and blackberry syrup. Add ice, cover, and shake well for about 15 seconds. Strain into a martini glass and top with prosecco.