The structural and magnetic properties of Fe_χMn_1−χ alloy thin films deposited on Cu(001) and on Co/Cu(001) are investigated. A layer-by-layer growth mode of the alloy films in a large range of concentration is deduced from the presence of regular oscillations in medium-energy electron-diffraction intensity. Low-energy electron-diffraction experiments reveal that Fe₅₀Mn₅₀ films keep the fcc structure when growing on Cu(001) and on Co/Cu(001). Magneto-optic Kerr effect is used to study the magnetic properties of Co/FeMn bilayers. A temperature-dependent increase in the coercive field of the Co film, when the thickness of the adjacent FeMn film exceeds a certain value, is related to the antiferromagnetic state of the Fe_χMn_1−χ alloy.