In a medium saucepan set over medium-high heat, stir together the cranberries, honey and water. Bring the mixture to a boil, then reduce heat slightly and cook until the cranberries are popped, 15 to 20 minutes. Cool to room temperature.
Pour the cranberries into a fine mesh sieve set over a bowl. Using a rubber spatula, push the cranberries through the sieve so that the syrup collects in the bowl. Discard the cranberry skins.
Spoon 2 tablespoons of the syrup into each of 6 champagne flutes. Top each with Prosecco. Serve.