In a saucepan, combine water, sugar, and corn syrup.
Bring to a boil and add jamaica flowers.
Remove from heat, cover, and steep for at least 20 minutes (or up to 2 hours).
Pour the mixture through a strainer, pressing firmly on the jamaica to extract as much liquid as possible. Add lime juice.
Pour mixture into an ice cream maker and freeze according to manufacturer’s directions.
If an ice cream maker is not available, pour a mixture into a 13×9 baking dish. Slide the dish into the freezer. Stir thoroughly after 45 minutes, then twice more at 15 minute intervals.