In a medium saucepan, stir together sugars, ice cream, butter, corn syrup, salt, and cinnamon. Bring to a boil over medium-high heat; reduce heat to medium-low, and simmer, stirring frequently, for 2 minutes. Remove from heat; stir in rum. Return to heat, and bring to a boil over medium-high heat; cook, stirring frequently, for 1 minute. Remove from heat, and pour in to a small heatproof bowl. Let cool for 15 minutes before using, stirring occasionally.