Watermelon-Lime Frozen Agua Fresca

Put 9 cups of the watermelon cubes on a rimmed baking sheet, wrap with plastic and freeze until solid, about 1 hour. Meanwhile, freeze 6 highball glasses. Chill the remaining 1 1/2 cups of watermelon. In a blender, combine half of the chilled watermelon with half of the frozen watermelon and half of the lime juice, sugar and rum. Blend until smooth. Pour into 3 of the frozen glasses. Repeat with the remaining chilled and frozen watermelon, lime juice, sugar and rum. Serve the agua fresca at once.