Place the lime juice and sugar in a small saucepan and heat on high, stirring, just until sugar is dissolved.
Pour the lime syrup into a blender, along with the coconut milk and cucumber. Blend until smooth. Run the mixture through a fine mesh sieve and discard solids.
Pour the liquid into popsicle molds and freeze for 1 hour. Insert popsicle sticks and freeze until solid, at least another 6-7 hours.
Recipe by The Kitchen McCabe at https://www.thekitchenmccabe.com/2015/06/16/cucumber-coconut-lime-popsicles/