Stir together the tart cherry juice and ginger in a saucepan (nonstick will work best), and cook over medium heat until combined. Reduce to medium-low, and simmer uncovered. You want the heat to be just high enough that the simmer is nice and steady.
Cook for 20-30 minutes (depending on temperature and surface area), or until the mixture is thickened and reduced by more than half. As it gets thicker, check it more frequently, to avoid burning. Remove the mixture from the heat and take out the ginger slice. The reduction will thicken more as it cools.
To each glass, add some ice, about 1/2 cup cold ginger ale, and about 2 tablespoons to taste of the tart cherry reduction. Enjoy cold.