We present magneto-optical Kerr effect measurements of epitaxial Co/Cu(001) films studied in situ as submonolayer quantities of Cu are deposited under ultrahigh-vacuum conditions. An enhancement of the magneto-optical response coincides with changes in the remanence behaviour, which are shown to be due to a variation in the ratio of the uniaxial and cubic in-plane magnetic anisotropy strengths. The data are consistent with the preferential growth of Cu atoms at atomic steps due to the substrate: these Cu `wires' reduce the step-induced uniaxial anisotropy and their electronic structure strongly affects the magneto-optical response.