Add sugar and water to a medium saucepan. Heat over medium-high heat until the sugar dissolves. Add the ½ ounce of mint to the saucepan and bring to a boil. Reduce to a simmer and cook for 5 minutes. Remove from heat, cover, and let sit for 10 minutes.
Once the 10 minutes have passed, remove the mint leaves with a slotted spoon. Place mixture into medium bowl. Add fresh lime juice, silver rum, and brown sugar. Stir together. Cover and refrigerate for 4 hours.
After the mixture has chilled, add lime zest and thinly sliced mint leaves. Freeze in ice cream machine according to manufacturer instructions. Note: because of the rum and all the sugar, the mixture will take a while to freeze. Mine took almost 25 minutes.
Transfer frozen mixture to storage container and freeze until firm, about 8 hours.