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The electronic structure and spin-dependent tunneling in epitaxial Fe/MgO/Fe(001) tunnel junctions are studied using first-principles calculations. For small MgO barrier thickness the minority-spin resonant bands at the two interfaces make a significant contribution to the tunneling conductance for the antiparallel magnetization, whereas these bands are, in practice, mismatched by disorder and/or small applied bias for the parallel magnetization. This explains the experimentally observed decrease in tunneling magnetoresistance (TMR) for thin MgO barriers. We predict that a monolayer of Ag epitaxially deposited at the interface between Fe and MgO suppresses tunneling through the interface band and may thus be used to enhance the TMR for thin barriers.