Combine mint leaves, sugar, and 1/4 cup water in a small pot. Bring to a boil and stir until the sugar has dissolved. Transfer the mixture to a heatproof container and chill, uncovered, until cool, about 30 minutes.
Strain mint syrup into a blender; discard mint leaves. Add chopped apricots and lime juice and blend until very smooth. Using a fine-mesh sieve, strain into a pitcher; discard solids. Add 2 cups water and stir well to combine. Squeeze a bit of fresh lime over each glass and garnish with mint sprigs. Enjoy!
Tip: If you can't find apricots at your local grocer, watermelon also works well.