In a medium saucepan over medium heat, combine sugar and water until the sugar dissolves. Add the mint sprigs; stir until mixture comes to a boil; reduce heat to low and simmer 5 minutes. Remove from heat, and let stand approximately 10 minutes.
Place a strainer over a large bowl and pour syrup mixture through (straining out the mint). Add lime juice, rum and lime zest to the strained syrup mixture and stir to combine.
Chill the mixture in a stainless steel or pyrex pan in the freezer. Stir with a fork every hour until frozen.
Serve with Mother Rucker's Sweets Lemon Cookie Brittle™.