A basic recipe that can be used for other berries as well. The amount of sugar syrup may need a little adjusting according to the sweetness of the fruit, but don't omit the lemon juice--it helps bring out the fullest fruit flavors.
Combine water and sugar in a medium saucepan and bring to a boil, stirring to dissolve the sugar. Boil for 2 minutes, then remove from the heat. Stir in the lemon juice and creme de cassis. Transfer to a medium bowl and let cool completely.
Puree the raspberries in a food processor, then strain through a fine-meshed sieve into the bowl containing the sugar syrup. Chill until cold, then process in an ice cream maker according to the manufacturer's directions. (Alternatively, freeze the mixture in a shallow metal cake pan or ice cube tray until almost solid. Break into chunks and process in a food processor.) Transfer the sorbet into a freezer container and freeze at least 1 hour.
Make Ahead Tip: The syrup (Step 1) can be made up to 3 days in advance and refrigerated. Freeze the sorbet for up to 1 day. Let soften slightly in the refrigerator for 15 to 20 minutes before serving.
1 fruit, 3 other carbohydrate