In a medium skillet, combine the mango, sugar, water, and cinnamon. Over medium high heat, bring the liquid to a boil. Lower the heat to medium and stir the mixture constantly for 3 to 4 minutes, until the sauce thickens slightly. In a small saucepan, warm the rum over low heat and stir in the cornstarch mixture until it thickens.
Transfer the sauce to a heatproof bowl. Bring the mango sauce and the pan of warm rum to the table and spoon ice cream into six individual bowls. Light the rum in the pan and pour it, still flaming, into the mango sauce. Spoon some of the mango sauce over each dish of ice cream.