Combine muddled mint, concentrate, water, sugar, rum and fresh lime juice in a large freezer-safe container, stirring until the ingredients are completely incorporated. Place in freezer overnight.
When you’re ready to serve, scoop ingredients into a cold glass, then add a splash of lemon-lime soda, stirring gently.
Garnish with fresh mint and a wedge of fresh lime.
Notes
Black rum is more rich in flavor than spiced or light rum—but don’t hesitate to use other varieties of rum, if that’s what you have on hand!