In a small saucepan, combine the sugar, water and mint. Bring to a boil, then lower the heat and cool until the sugar is dissolved, stirring often. Remove from the heat, cover and let steep for 10 minutes.
Strain the mixture into a large bowl, preferably one with a spout for easy pouring. Stir in the rum, lime juice and zest, and divide between ten 3-ounce Popsicle molds.
Freeze until solid, 6 to 8 hours, then unmold and serve.