The invention relates to an at least two-stage method for corrosion protection treatment of metal surfaces, wherein, in a first step (i), an organic coating comprising an aqueous phase (A) is applied to the metal surface and, in a following step (ii), the organic coating applied to the metal surface contacted with an acidic, aqueous composition (B) comprising at least one or more water-soluble compounds containing at least one atom selected from the elements Zr, Ti, Si, Hf, V and/or Ce and one or more water-soluble compounds that release copper ions. The invention further comprises a metal component that is at least partially made of steel, iron, zinc and/or aluminum and the alloys thereof and has been treated by the method according to the invention, and to the use thereof in automobile construction and in the construction industry and for producing household appliances and electronics housings.