1. Make a punch bowl out of the hollowed watermelon by cutting off approximately the top fifth of the watermelon (or just enough to open up a diameter of a bowl -- leave as deep a bowl as possible). Also cut thin slice off the peel and rind on bottom (but not into the flesh) to create a flat surface for the punch bowl to stand.
2. Scoop out flesh from the melon. Working in batches, puree about half the watermelon flesh in a food processor to make 1.25 L (5 cups) juice. Strain juice and discard solids (reserve remaining watermelon flesh for another use). Mix together melon juice, rum, lime juice, sugar, and mint leaves in a pitcher. Refrigerate for 1 hour.
3. Before serving, pour punch into watermelon bowl. Add sparkling wine and stir. To serve, add ice to tall glasses and fill with punch.

Chef’s secret

If you like, garnish glasses with lime wheels.