Remove the rind and cut the watermelon into small pieces. Using a blender, finely puree the melon. Then, strain the puree through a fine-mesh sieve to remove any remaining pulp.
Now, mix the watermelon juice with the Aperol, Prosecco, and sparkling water. Combine well, but carefull not to shake/stir too vigorously. Otherwise the drink may go flat.
Wash the mint leaves and shake them dry. Halve the lime and cut one of the halves into slices.
Divide the ice cubes among four glasses and pour the cocktail evenly over them. Garnish with mint leaves, lime slices, and drizzle some juice from the other lime half into each glass.