Heat the heavy cream and half and half in a stainless-steel saucepan. Beat yolks and sugar together. Slowly add cream to egg mixture, whisking constantly. Return mixture to saucepan and reheat. Stir with a wooden spoon, scraping bottom of saucepan, until mixture thickens and coats back of spoon. Remove from heat and strain. Stir in black-currant puree.
Chill custard completely. Pour into an ice-cream maker and freeze according to manufacturer's instructions.