Place sugar, 2 sprigs of rosemary and 1 cup of water in a small saucepan, and stir over low heat until the sugar dissolves. Remove from the heat and let the syrup sit for 1 hour. Remove the rosemary sprigs and store the syrup in a glass jar in the fridge for up to 1 month.

Combine the rosemary simple syrup, rum and lime juice in a shaker. Add ice, shake and strain into a glass over ice. Top with soda and garnish with a rosemary sprig and wedge of lime.