Add the sugar and water to a medium saucepan and stir to combine. Bring to a simmer over medium heat, stirring occasionally. Turn heat down to a low simmer and continue to simmer and stir until sugar is completely dissolved. Remove from heat and allow to cool to room temperature.

Add the cooled sugar syrup to the pitcher of a blender along with the 1 cup of roughly chopped pineapple and the lime juice. Blend until smooth, about 30 seconds on high. Once the mixture is smooth, stir in the finely chopped pineapple and lime zest.

Pour mixture into 10 popsicle molds, insert a stick in each one, and freeze until solid, at least 6 hours. When ready to serve, run molds under hot water to make popsicles easier to remove.