In a small saucepan, make a simple syrup by adding 1 cup of water and the cane sugar, and put over medium heat. Whisk until it the sugar is completely dissolved, remove from the heat.
In a pitcher, add the simple syrup, remaining 4 cups water, and 2 cups lemon juice. Stir to combine completely.
Fill popsicles molds, place in sticks, and freezer overnight.