In a small saucepan combine sugar and water. Heat, stirring constantly, over medium heat until sugar fully dissolves. Take off heat, stir in mint, cover, and let infuse for 20 minutes.
While mint syrup infuses, in a blender puree watermelon. Strain through a fine-mesh sieve, then line the sieve with a piece of thin kitchen towel (such as a flour sack towel) or several layers of cheesecloth, and strain the juice again. Set aside.
Strain the mint syrup and stir ⅓ cup of the syrup into the watermelon juice. Stir in lime juice.
For each serving, fill a tall glass with ice. Pour in ½ cup watermelon mixture, then top off with ⅓ cup sparkling water.
Garnish with watermelon wedges and mint sprigs.