Excellent adhesion to subsequently applied paint or elastomer is obtained by coating a metal substrate with a phosphate conversion coating by contact with a liquid phosphating solution that contains zinc cations, phosphate anions, and at least one adhesion promoter selected from (i) film-forming organic substances, (ii) polymers of vinyl phenols modified by substitution of substituted aminomethyl moieties on their aromatic rings, (iii) inorganic oxides of one of the elements silicon, aluminum, titanium, and zirconium. Preferably, the phosphating solution also contains manganese and nickel cations and either iron cations or hydroxylamine. If adhesion to paint is desired, the adhesion promoter preferably is an acrylic film-forming substance, while if adhesion to elastomers is desired, the adhesion promoter preferably is a polymer of vinyl phenol and the phosphating solution preferably also contains calcium cations.