Make a syrup by bringing the sugar and water to a boil in a small saucepan over high heat, stir and boil until the liquid becomes clear, about 1 minute. Set aside to cool.
Blend the drained pineapple in a blender or food processor until very smooth and frothy. In a large bowl, whisk together the syrup, pineapple puree, coconut milk, and lime juice. Refrigerate until chilled, about 3 hours.
Freeze in the freezer canister of an ice cream maker according to the manufacturer's directions.
