In a blender, combine the blackberries and water. Blend until pureed. Strain the juice thru a fine mesh sieve and discard all the seeds.

In a glass, add the mint leaves, juice from half a lime and the lime rind, and a splash of the soda water. Gently muddle to release the flavor from the mint leaves. Fill the glass with ice and add the rum, blackberry juice, and top with soda water. Stir to combine. Garnish with lime wedge, mint sprig and blackberries.