1In 1-quart saucepan, mix mint syrup ingredients. Heat to boiling over high heat, stirring occasionally. Reduce heat to low; simmer 5 minutes or until sugar is dissolved. Remove from heat; let stand at room temperature about 20 minutes.
2Strain mint leaves mixture into pitcher; do not push on leaves or syrup will turn a dark green color. Add rum, lime juice and club soda to pitcher.
3Fill 4 low-ball glasses with ice cubes. Place halved lime slices in glasses, pushing down along side of glass, allowing ice cubes to hold them in place.
4Pour mint mixture into glasses over ice. Garnish each with mint sprig.