Wash and chop the limes. Wash the mint and pluck off all but a little. Crush both with sugar, possibly with a wooden pestle. Leave to stand. Cut a lid off each melon, remove seeds.
Cut melons flat at the bottom. Cut out balls, hollow out melons. Put mint and about half of the balls on skewers. Fill the remaining balls and ice cream 2/3 high into the melons. Sieve the mojito, fill with rum and mineral water.
Place each on a plate with ice using skewers. Decorate with remaining mint and serve with drinking straws.