The in-plane magnetic anisotropy of ultra-thin Co films, epitaxially grown on Cu(1 1 17), was determined in situ by means of the magneto-optic Kerr effect down to thicknesses as low as 2 monolayers. Uniaxial and biaxial anisotropy contributions were observed. At room temperature, the uniaxial component is dominant and the easy axis of magnetization is parallel to the step edges. Above 4 monolayers the magnetic anisotropy exhibits a thickness dependence which can be described by volume and interface contributions. For thinner films a pronounced deviation from that behavior is found. The anisotropy drops abruptly by one order of magnitude below 3 monolayers. Thickness dependent relaxations are proposed as driving forces for that behavior. (C) 1996 American Institute of Physics.