In a saucepan, bring the water and sugar to a boil over medium-high heat.  Boil for 1 minute, then remove from heat and allow mixture to cool for 10 minutes (or more).

Then add the sugar/water mixture, basil leaves and lime juice to a blender.  And carefully (use caution when blending warm liquids) puree the mixture for about 1 minute, or until well-blended..  Pour through fine-meshed strainer, and stir in the lime zest.

Freeze immediately in an ice cream maker according to manufacturer’s instructions until slushy-firm.  Scoop into a storage container and freeze until firm enough to scoop.