Combine cranberry juice and ginger in a 2- to 3-quart pan over medium heat. Bring to a simmer and cook to infuse flavor, about 20 minutes.
Stir in lime juice and 1/3 cup sugar; stir until sugar has dissolved. Taste and add up to 1/4 cup more sugar if desired.
Pour through a fine strainer into a thermos or heatproof pitcher (discard solids). Serve hot.