For Simple Syrup: Put the sugar and water in a small saucepan. Cook over low heat, stirring, until the sugar dissolves. (Alternatively, heat in the microwave.) Store in a sealed container in the refrigerator.
For Old Cuban: Muddle the mint leaves with the rum, lime juice, sugar or simple syrup, and bitters in a cocktail shaker.
Add ice, cover, and shake vigorously, or stir, until combined and chilled, about 30 seconds. (In general, the drink is ready when the shaker mists up.) Pour the muddled liquid into a chilled champagne flute, holding back the ice, but letting some of the mint leaves fall into the glass.
Add enough Champagne to almost fill the glass to the top. Garnish with a sprig of mint. Enjoy.