Add the pomegranate juice to a small saucepan over low heat.
When the juice is warm, add the sugar; stir until sugar is completely dissolved.
Remove pan from heat and refrigerate until cold or overnight.
Add the lime juice to the chilled mixture; stir well.
Freeze mixture in your ice cream maker by following the manufacturer’s directions.
The sorbet will be soft but ready to eat; for firmer sorbet, transfer to a freezer-safe container and freeze several hours.
*Pomegranate Cinnamon Sorbet-add two 4-inch cinnamon sticks to the pan with the sugar; proceed with the recipe as directed, removing the cinnamon sticks before freezing.
*Pomegranate Orange Sorbet-substitute 1 ½ cup orange juice for 1 ½ cup pomegranate juice; proceed with the recipe as directed, adding 1 teaspoon freshly grated orange zest before freezing.
*Russian Pomegranate Sorbet-Add ½ cup vodka before freezing; proceed with the recipe as directed.