Make the syrup: In a small saucepan over medium heat, place mint leaves, sugar, and water. Stir to dissolve. Bring to a simmer, watching carefully to keep mixture from boiling. As soon as steam and a few bubbles appear, remove from heat, cover, and steep 30 minutes. Strain into a small pitcher, jar, or bowl. Discard leaves, and chill mint syrup.
To make the soda: Place ice in a tall glass. Add 1/4 cup cooled mint syrup and top off with club soda. Stir gently to combine, garnish with mint sprig, and serve.
Syrup can be refrigerated in a covered container up to 1 week.