Combine all the ingredients in a 2-quart saucepan over medium-high heat and bring to a boil, stirring often. Once the mixture comes to a boil, reduce the heat to a medium-low and allow the pot to simmer until the fruit is tender, about 10 to 15 minutes. Once the fruit is cooked, remove from the heat, and allow to cool slightly. Working in batches, puree the mixture in a blender*. Chill thoroughly before proceeding.
When the mixture is thoroughly chilled, transfer to an ice cream maker and process according to the manufacturer's instructions.
*When blending hot liquids: Remove liquid from the heat and allow to cool for at least 5 minutes. Transfer liquid to a blender or food processor and fill it no more than halfway. If using a blender, release one corner of the lid. This prevents the vacuum effect that creates heat explosions. Place a towel over the top of the machine, pulse a few times then process on high speed until smooth.