Step 1Cut the melon in half and scoop out the seeds. Using a spoon, scoop out chunks of melon flesh, putting them in a food processor work bowl. Don't dig too deep -- the melon close to the peel has a strong cucumber flavor.
Step 2Puree the melon and stir in 2 tablespoons sugar. Taste and add one more tablespoon if necessary. Because chilling reduces flavor, the mixture should be very sweet.
Step 3Pour the puree into a 7- by 11-inch glass baking dish (or another dish that will hold the puree to a depth of three-fourths to 1 inch). Freeze the puree for 30 minutes. Remove the pan from the freezer and stir the puree with a fork, breaking up any chunks of ice. Repeat 4 or 5 times over 2 to 3 hours. Each time, the ice will be a little less liquid and will stick together more. When it is firm enough to hold a shape, it is done.
Step 4Try not to let the melon ice freeze solid. If it does, chop it into small pieces in the pan and grind it in the food processor.(The result will be lighter and fluffier and the flavor will not be as dense and luscious.)
Step 5Stir together the berries with 1 tablespoon sugar and the Port. Let the fruit mixture sit for 10 to 15 minutes. To serve, spoon the melon ice into martini or short wine glasses and spoon some of the berries with their liqueur over the top of each.