In a medium saucepan, combine sweetened condensed milk and cocoa. Cook over low heat, stirring constantly, until the mixture is smooth and starts to thicken (about 10 minutes). Remove the mixture from the heat and cool slightly.
Gradually add heavy cream, light cream and vanilla to the mixture, beating with a whisk until well blended; refrigerate the mixture until cold (about 3 hours).
Using an ice cream maker, freeze the mixture according to manufacturer's instructions.