Add sugar and water to a medium saucepan. Bring the mixture to a simmer over medium heat, stirring occasionally. Allow it to simmer until the sugar is completely dissolved, about 5 minutes. Do not use high heat - or boil the mixture - as you risk scorching the sugar.
Remove the mixture from the heat, and allow it to cool for 10 minutes. Then stir in the lilikoi juice.
Cover with plastic wrap and refrigerate until completely chilled, at least 4 hours - or even better, overnight.
Place the container you plan on storing the finished sorbet in, into the freezer. This way, it is ice-cold when you add the finished sorbet to the container and won't melt the sorbet.
After the mixture has completely chilled, pour the mixture into an ice cream maker and follow the manufacturer's instructions. As a guide, I run my ice cream maker for 20-25 minutes. The sorbet should be thick and fluffy but not watery.
Add the sorbet to the chilled container, and freeze for one to two hours before serving to allow the sorbet to firm up.