2. Make a syrup by combining the water, sugar and cinnamon stick in a saucepan and bring to a boil.
3. Meanwhile, puree the berries and strain out the seeds with a fine mesh strainer. Use a rubber spatula to push through as much puree as you can. You should have about 2 cups of raspberry puree.
4. Whisk the strained puree and lemon juice together in another bowl.
5. Let the syrup cool slightly, then remove the cinnamon stick and whisk half the syrup into the fruit puree and taste. Slowly add more syrup, tasting frequently. The mixture should taste distinctively tart and sweet; you might not use all the syrup.
6. Chill overnight or freeze right away in an ice-cream maker, then transfer to the frozen container and keep frozen until ready to serve.