Prepare an ice bath (ice and water) in a large bowl and set a medium size bowl in the bath.
Add sugar and water to a small saucepan and warm over medium high heat, stirring until sugar has completely dissolved. Bring the mixture to a boil and let bubble for 1 minute. Remove from the heat and pour into the medium bowl sitting in the ice bath. Stir occasionally to expedite the cooling process.
Once cooled, add meyer lemon juice and lemon zest. Once the mixture has completely chilled, cover and transfer for the refrigerator. Let the mixture sit for a few hours or up to a day in the refrigerator. The longer the mixture sits, the more tart it will be (from the lemon zest).
Once the desired tartness has been achieved, strain the mixture through a fine mesh basket strainer into a clean bowl or measuring cup. Discard lemon zest.
Place the mixture in an ice cream maker and freeze according to the manufacturer’s instructions.