Process for the anodic electro-dipcoating of conducting surfaces by dipping in an aqueous anodic electro-dipcoating bath and connecting up as the anode, in which an aqueous anodic electro-dipcoating bath is used that contains 1 to 15 wt. %, referred to the binder solids of the electro-dipcoating bath, of one or more phosphoric acid epoxy esters and/or phosphonic acid epoxy esters that have been produced in the presence of alcohols.