Put the sugar and water into a saucepan over medium-high heat and bring to a boil. When the sugar has dissolved, remove from the heat and let cool.
Put the mint leaves into a blender along with the lime juice, rum, and cooled sugar syrup. Blend for about 30 to 45 seconds or until the mint is finely chopped. Pour this liquid into a 9 by 13-inch metal cake pan and put it into the freezer. Scrape the mixture every hour with a fork to break up the ice crystals. After about 4 to 6 hours, depending on your freezer, the pan will be full of mojito ice flakes and will be done. Serve in small, cold bowls or glasses, garnished with reserved mint leaves.