Special equipment: Four 4-ounce ice-pop molds or six 3-ounce molds, 4 or 6 ice-pop sticks
Bring the sugar and 1/4 cup water to a simmer in a small saucepan. Cook over medium heat, stirring occasionally, until the sugar is completely melted, about 5 minutes. Turn off the heat, add the cilantro and stir until completely wilted. Let cool completely at room temperature, at least 30 minutes.
Put the melon, lime zest and juice and a pinch of salt in a blender. Strain the cilantro syrup into the blender, pressing the cilantro in the strainer with the back of a spoon to extract as much flavor as possible; discard the cilantro. Puree the melon mixture until completely smooth, scraping down the sides of the blender partway through if needed. Transfer the mixture to a large liquid measuring cup with a pouring spout.
Pour the mixture into the molds, leaving 1/4 inch of space at the top (the mixture will expand). Insert the sticks. Freeze until solid, 5 hours to overnight. Unmold, and enjoy immediately.
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