In a medium microwave-safe bowl, combine the water and sugar. Microwave for 1 minute, stir, then microwave for 1 more minute and stir to create a syrup. Cover and refrigerate at least 2 hours.
Pour the pineapple juice into an ice cream machine and add ½ cup chilled sugar syrup. Add the lime juice. And then add the ice cream too. Follow the ice cream machine instructions and run about 20 minutes. Churn until creamy.
Serve immediately, or transfer to a large plastic container, cover and freeze overnight for a harder consistency.