Place the sugar, melon and lime juice in a blender. Process the mixture until smooth. Transfer the mixture to a 9-by-13-inch baking dish.
Freeze for 2 or 3 hours, and then begin the process of lightly scraping the top frozen layer with a fork.
Return the pan to the freezer with the shaved ice on top. Remove a couple of hours later and continue scraping with a fork. Repeat the occasional scraping process until the entire mixture is shaved.
Store, covered in plastic wrap, until serving.
Serve in pretty glasses with mint.