Prepare the glass. Place the blackberries, mint leaves, sugar, and lime juice into the tall glass you intend to use for serving.
Muddle the ingredients. Using a muddler, gently press down and twist on the blackberries, mint leaves, sugar, and lime juice mixture until the mint is slightly bruised and the blackberries release their juices. This usually takes about 30 to 45 seconds. Avoid over-muddling to prevent the mint from becoming bitter.
Add ice. Fill the glass about three-quarters full with ice cubes to chill the mixture and refresh the drink.
Pour in the liquids. Slowly add the white rum and club soda to the glass, pouring over the ice to mix the drink gently and maintain its fizz.
Stir and garnish. Stir the mixture gently with a long spoon to combine all the ingredients well. Garnish with a few whole blackberries, a sprig of fresh mint, and a lime wedge on the rim of the glass.
Serve immediately. Serve the mojito immediately to enjoy its freshness and vibrant flavors.