Cut off a lid from each melon. Remove the seeds from the melons with a spoon. Use a ball cutter to cut out balls from the melon pulp and place them on a plate. Wash the mint, dab dry and pluck off the leaves except for something to decorate. Put half of the melon balls and some mint leaves on 4 skewers.
Wash and quarter the limes and crush them with brown sugar and about 12 mint leaves. Pour through a sieve, add rum and spread with some crushed ice and the remaining melon balls on the 4 melons. Top up with mineral water. Arrange each melon on a plate with crushed ice and a melon skewer. Stick straws into the melons and serve immediately decorated with remaining mint