CoO/MgO/Fe/Ag(001) films were grown epitaxially and studied using Magneto-Optic Kerr Effect and X-ray Magnetic Linear Dichroism (XMLD). For thick CoO films we show that both the Fe coercivity and exchange bias exhibit the expected monotonic decrease with increasing MgO thickness. For thin CoO films, however, we find that while the coercivity decreases monotonically, the exchange bias increases initially and then decreases with increasing MgO thickness. By measuring the CoO XMLD as a function of MgO thickness, we show that the unusual behavior of the exchange bias at thinner CoO is due to a transition of the CoO spins from rotatable to frozen spins as the MgO thickness increases.