In a bowl, stir together the cream and milk. Add the sugar substitute and whisk until the sugar substitute is dissolved, 3 to 4 minutes. Test for graininess by tasting a small amount of the liquid; it should feel smooth on the tongue and there should be no sugar substitute visible on the bottom of the bowl when it is stirred or spooned out. Continue whisking, if necessary, to ensure that the texture of the finished ice cream will be smooth. Stir in the vanilla.
Place a piece of plastic wrap directly onto the surface and on top of the bowl. Refrigerate for at least 3 hours or up to 24 hours.
Prepare an ice cream maker with at least a 1-quart capacity according to the manufacturer's instructions. Remove the plastic wrap from the cream mixture and bowl. Pour the well-chilled cream mixture into the mixing container of the ice cream maker and freeze according to the manufacturer's instructions. The timing will depend on the type of machine and the temperature of the cream mixture.