Place the watermelon in a single layer on a rimmed baking sheet, and freeze until completely frozen, like ice cubes, about 2 hours.
Meanwhile, cook the sugar and 1/2 cup water in a small saucepan over medium heat, stirring occasionally, until the sugar is completely melted, about 5 minutes. Turn off the heat, add the chopped basil and stir until completely wilted. Cool completely at room temperature, at least 1 hour.
When the watermelon is completely frozen, put half in a blender. Strain the basil syrup into the blender, use the back of a wooden spoon or spatula to press down on the basil to extract as much flavor as possible, and discard the basil. Add the lemon juice, and blend until the mixture is mostly smooth, pushing it down with a wooden spoon or spatula if needed. Add the remaining watermelon, and continue to blend until the mixture is completely smooth.
Spoon the mixture into 4 short glasses (or wineglasses, if you want to be fancy), and garnish each with a basil leaf.