Melt chocolate over double boiler over low heat. Once it has fully melted, turn off the heat and allow to cool slightly at room temperature.
Beat cream cheese in a mixing bowl until it is soft and smooth, then slowly add the sugar, beating after each addition. Next, beat in the sour cream, chocolate, and heavy cream.
Add the vanilla extract and lemon juice to the cream mixture and beat until thick and smooth.
Cover and chill in the refrigerator for 2 to 3 hours before making ice cream.
When the mixture has cooled, pour it into a crank freezer or electric ice-cream maker and follow the manufacturer’s directions.
If you do not have an ice cream maker, you can improvise by filling a quart-size zipper-style bag two-thirds full with the liquid ice cream mixture and zipping it closed, then putting it into a gallon-sized zipper-style bag.
Hold the mouths of the bags flush while somebody else fills the gallon-sized bag with crushed ice and rock salt, in layers. Zip the gallon-sized bag closed and massage and shake the bags to achieve the same effect as a classic ice cream freezer. You may have to do this in a few small batches. Keep checking the quart bag to see if your ice cream has frozen sufficiently.
Pack ice cream into a 1-quart lidded container. Cover with plastic wrap, pressing it onto the surface of the ice cream. Place the ice cream into the freezer and leave until fully set, about 5 to 6 hours or overnight.