Make a punch bowl out of the hollowed watermelon by cutting off approx. the top fifth of the watermelon (or just enough to open up a diameter of a bowl - leave as deep a bowl as possible with the rest of the "shell"); 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.
Scoop out flesh from the larger portion of watermelon (reserve removed top portion for another use). Working in batches, puree about half the watermelon flesh in a food processor to make 5 cups (1.25 L) juice; strain and discard solids (reserve remaining watermelon flesh for another use). In pitcher, mix watermelon juice with rum, lime juice, icing sugar and mint leaves. Chill 1 hr.
Before serving, pour punch mixture into watermelon punch bowl. Top with sparkling wine. To serve, add ice to tall glasses and ladle in punch from watermelon.
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