In a small saucepan, combine water and sugar. Heat over medium heat, stirring until the sugar is completely dissolved. This will be your simple syrup.
Let the simple syrup cool to room temperature.
Peel and cube the pineapple. Puree in a blender until smooth.
Combine the pineapple puree with the cooled simple syrup. Add the freshly squeezed lime juice to enhance the pineapple's sweetness.
Pour the mixture into an ice cream maker and churn according to the manufacturer's instructions.
If you don't have an ice cream maker, pour the mixture into a shallow container and place it in the freezer. Every 30 minutes, stir with a fork to break up ice crystals. Repeat this process for 3-4 hours.
Before serving, let it sit at room temperature for a few minutes to soften slightly for easier scooping.
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