Make the blackberry syrup: Combine the blackberries, sugar and 1/2 cup water in a small saucepan over medium-high heat. Bring to a boil, then reduce the heat and simmer until the syrup is dark red, about 8 minutes. Remove from the heat and stir in the basil sprig; let cool. (Makes 10 ounces; refrigerate up to 1 week.)
Combine the vodka, lime juice and 1 ounce blackberry syrup in a cocktail shaker; fill with ice. Shake well and strain into an ice-filled glass. Garnish with blackberries and basil.