We have investigated the magnetic coupling between a metallic ferromagnet and an oxidic antiferromagnet in epitaxial single-crystalline Fe/CoO bilayers on Ag(001) using x-ray absorption spectroscopy. Absorption spectra taken from bilayers with different amounts of deposited Fe show only a weak indication for the formation of Fe oxide at the Fe/CoO interface. From the spectral shape, it is concluded that an FeO type of oxide is formed. X-ray magnetic circular dichroism (XMCD) measurements exhibit a sizable induced ferromagnetic signal at the Co L_2,3 absorption edges, corresponding to an interface layer of 1.1 ML of CoO in which the Co magnetic moments couple with the Fe moments. The angular dependence of the Fe XMCD and Co x-ray magnetic linear dichroic signals at the L _2,3 edges shows that the orientation of the Co and Fe spins is parallel along the crystallographic <110> directions.