Place the blackberries, mint leaves and simple syrup in a cocktail shaker and muddle with a cocktail muddler.
Add the lime juice and the rum.
Fill the shaker with ice.
Shake and strain into a glass filled with ice.
Top with the club soda.
Garnish with extra blackberry and mint leaves.
For the simple syrup, in a small saucepan, boil sugar and water over medium high heat until sugar dissolves.
Remove from heat and let cool.