1Put the sugar and water in a saucepan. Stir over medium heat until the sugar has dissolved.
2Add the raspberries. Increase the heat to medium high and bring the syrup to a simmer. Simmer for 2 minutes. Set aside to cool for 15 minutes.
3Pour the raspberry mixture into a blender. Blend until smooth.
4Strain the mixture into a bowl, pushing the mixture through the sieve. Discard the solids.
5Stir the lime juice into the raspberry syrup. Cover and refrigerate for 3 hours until cold.
6Pour the chilled mixture into the ice cream bowl. Set the ice cream maker to the desired setting and churn.
7Once the sorbet is frozen, spoon or scoop into chilled bowls or glasses and serve or transfer to a freezer-safe container, just large enough to contain the sorbet. Freeze for 2-3 hours or overnight for firmer sorbet.
8Store sorbet in the freezer for up to 3 weeks. The sorbet will become very firm after freezing overnight, so remove from the freezer and stand at room temperature for 5-10 minutes before serving for easier scooping.