Thanks to watermelon's natural sweetness, this drink doesn't call for sugar.
Puree watermelon in a blender until smooth. Strain mixture through a fine sieve into a pitcher. Stir in lemon juice, lemongrass, ginger, and 1/8 teaspoon salt. Refrigerate until chilled, about 20 minutes.
Divide juice among 6 ice-filled glasses, and garnish with lemongrass stalks.
Juice can be refrigerated for up to 3 days.