Cut rind off watermelon and discard. Cut fruit into 1/2-inch chunks.
Line a 10- by 15-inch pan with plastic wrap. Arrange fruit on wrap in a single layer. Freeze until firm, 1 to 2 hours; if making ahead, transfer frozen chunks to a freezer bag.
In a blender, combine half the pineapple juice, rum, and lemon juice. Whirl, gradually dropping in half the frozen watermelon chunks, until smoothly puréed. Sweeten to taste with sugar and pour into a pitcher or directly into glasses. Repeat to make a second batch.
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* The % Daily Value (DV) tells you how much a nutrient in a serving of food contributes to a daily diet. 2,000 calories a day is used for general nutrition advice.