In a small saucepan heat and stir the water, sugar, and mint over medium heat just until simmering, stirring to dissolve sugar. Remove from heat. Transfer to a medium bowl. Cover and cool in the refrigerator for at least 1 hour.
Strain mixture through a fine-mesh sieve; discard mint leaves. Stir in lime juice. Cover and chill for at least 4 hours or up to 24 hours.
In a 1-quart ice cream freezer, freeze chilled mixture according to manufacturer's directions for 10 minutes. Add rum; continue freezing according to manufacturer's directions. Transfer to a freezer container. Cover and ripen in the freezer for at least 4 hours.