For mint sorbet, pour the water into a medium sauce pan. Add the sugar and dissolve. Add the lime and lemon zest. Bring the mixture to a boil.
At this point, for rosemary sorbet add the rosemary sprigs to the hot syrup and let steep.
Allow the mixture to cool to room temperature and then remove the rosemary sprigs.
Add the lime and lemon juice, the minced mint and the rum. Transfer to an airtight container and refrigerate for at least 4 hours.
Place the syrup into an ice cream maker and follow manufacturer's directions. When the sorbet has frozen (approx. 35-40 min.) transfer back to the airtight container and place in the freezer for at least 2 hours to set.