Cut a little off of one end of the watermelon to create a sturdy base for it to stand upright. Cut off about 1 inch from the top of the watermelon. Insert a stick blender into the top of the watermelon and blend, scraping down the sides so all the flesh can be blended.
Remove half of the watermelon puree and freeze it as ice cubes or save for a non-alcoholic beverage. Pour the vodka into the watermelon bowl and stir. Serve the punch over ice and garnish each glass with a sprig of mint.