Add the cherries to a saucepan with the sugar and pomegranate juice. Bring to a slow boil and cook until the cherries are soft and the liquid is syrupy, 15 to 20 minutes. Cool to room temperature.
Add the cherries and juice to a blender or food processor. Blend until smooth, stopping short of totally liquefying it if you'd like a little pulp. Add the milk, cream and lemon juice and blend briefly.
Freeze according to your ice cream maker's instructions, and then transfer to an airtight freezer container and freeze for at least 24 hours before serving.
Scoop into a pretty glass and serve. If you're feeling dangerous, mix with a little cold white wine for a delicious "milkshake." Just don't tell anyone I told you to do that.