Pour the ginger beer out in a large bowl and set aside. (The idea here is to let some of the carbonation fizz out of the popsicle so it doesn't overly expand while freezing.)

To make the ginger simple syrup, combine the sugar, water, and ginger slices in a small saucepan over medium heat. Stir until sugar dissolves, then set aside to steep for 10 minutes. Strain into a jar with a tight fitting lid.

To make the popsicles, combine the reserved ginger beer, 1/4 cup ginger simple syrup, dark rum, and lime juice, and pour into popsicle molds, leaving about 1/2-inch of space at the top to allow for lime addition and expansion as it freezes. Push a slice of lime into each mold, set sticks in place, and freeze. Since they're boozy, popsicles will cure best when frozen overnight.