Make the Lime-Basil Syrup ahead so it cools down and is ready when you are. Prep: 3 minutes.
Place first 3 ingredients in a sturdy old-fashioned glass. Crush to a pulp with a muddler or wooden spoon.
Add lime juice, Lime-Basil Syrup, and cracked ice. Stir in rum, and top with club soda.
Small Bites, Big Nights (Clarkson Potter, 2007; Table 8, South Beach and New York