Blend the honeydew with the lime juice until completely smooth, then strain through a fine sieve, pressing on the pulp. You should have about 3 ounces of juice.
Press the mint leaves gently with the simple syrup in the bottom of a stemless wine glass, just enough to release their oil.
Fill the glass with crushed ice and pour in the honeydew juice.
Top with the sparkling water and stir once.
Stand the skewer of honeydew balls and the mint sprig in the ice and serve right away.
Good to Know
Taste the strained juice before adding the syrup: a very ripe melon may need none at all.