Bring 1 cup Champagne and the sugar to a boil in a small saucepan. Remove from heat; let cool completely. Stir together gelatin and remaining cup Champagne in a small bowl; let stand until softened, 2 to 3 minutes. Stir in sugar mixture.
Divide Champagne mixture among 6 (3 1/2-inch-long, 1 1/2-inch deep) oval molds or six (4-ounce) ramekins. Tap firmly on counter to release any air bubbles. Place 1 sprig currants in each ramekin. Refrigerate until firm, 1 to 1 1/2 hours. To unmold, set bottom of each ramekin in hot water about 10 seconds to loosen; invert onto a plate.
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