The properties of the magnetic domain structures of ultrathin fcc cobalt films epitaxially grown on Cu (001) have been examined using an ultrahigh vacuum surface magneto-optic Kerr effect instrument. The evolution of magnetic behavior is observed for film thicknesses ranging from 1.4 to 7.5 monolayers. The coercivity is sensitive to film growth temperature and thermal cycling history. The coercivity decreases with diminishing film thickness and falls to very low values for the thinnest layers. The results are discussed in terms of Neel domain-wall micromagnetics for ultrathin films.