Quarter the watermelon, then cut the flesh from the rind and cut the flesh into 1-inch pieces; you should have about 8 cups (3 pounds). Scoop the watermelon into a blender along with 1 1/2 cups water and process to a smooth puree (do this in batches if all of it won't fit into your blender).
Pour into a very large (1 gallon) pitcher or Mexican glass agua barrel. Add 4 cups water, along with the lime juice and sugar. Stir until the sugar dissolves, then give it a taste, adding more lime or sugar as you see fit. Serve over ice.