Place half of the watermelon cubes in a blender and process until smooth. Strain through a fine-mesh sieve set over a large bowl; discard the solids. Repeat with the remaining watermelon cubes. You should end up with about 1 quart of watermelon juice.
Add the lime juice and sugar to the watermelon juice, and stir until the sugar has dissolved. Taste, and add more sugar if necessary. Transfer the limeade to a nonreactive pitcher and refrigerate until thoroughly chilled.
Serve over ice in tumblers, with lime slices for garnish.
Yield: 5 cups, 4 to 6 servings