Set the flat side of a watermelon on a cutting board. Jiggle the melon to see if it will stay level. If not, slice a thin layer of rind off the bottom to make a flat surface. Then cut a large oval opening in the top of the watermelon.
Use a large metal spoon to scoop the flesh out of the watermelon, creating a punch bowl. Set the fruit chunks in a large bowl to collect the juices.
Working in two batches, place half the watermelon flesh into the blender. Add half of the rum, juices and sugar. Puree until smooth. Pour the punch into the watermelon bowl and repeat with the remaining ingredients.
Chill the punch until ready to serve. (Depending on the size of watermelon you may have extra in the blender pitcher for refilling the punch bowl.)
Some seedless watermelons have less baby seeds than others. You can strain the punch if you like, but I prefer the frothy texture of the unstrained punch.