Step 1Peel the pineapple and remove the hard core. Cut 1 whole slice of pineapple, about 1/4 inch thick. Slice it into 1/2-inch pieces and set them aside for garnish. Cut the rest of the pineapple into chunks and puree them in a blender. You should have about 2 cups. Put the puree through a strainer and discard the solids. Add sugar and rum to the blender and mix until the sugar is dissolved.
Step 2Using a melon-baller, scoop out most of the flesh of the honeydew and cantaloupe into little balls. Set aside.
Step 3Pour 1/3 cup of the fruit soup into each of 6 chilled glass dessert bowls. Add the pineapple, melon balls and blueberries to the bowls. Garnish with mint and serve immediately.