In a large bowl, stir together heavy cream, whole milk, and granulated sugar. Stir until sugar has dissolved.
Stir in pomegranate juice.
Put mixture in fridge to chill for at least 2 hours (I like to chill overnight if possible).
Once chilled, pour into ice cream maker and process according to manufacturer's directions, likely running for about 20 minutes. When ice cream is almost done processing, add in pomegranate arils.
Transfer ice cream to an airtight container and freeze for at least 4 hours before serving.
Serve with more pomegranate arils on top, if desired.