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High-quality cross sections for the reaction gamma d -> pi(0)d have been measured using the CLAS at Jefferson Lab over a wide energy range near and above the eta-meson photoproduction threshold. At backward c.m. angles for the outgoing pions, we observe a resonance-like structure near E-gamma = 700 MeV. Our model analysis shows that it can be explained by eta excitation in the intermediate state. The effect is the result of the contribution of the N(1535) S-11-resonance to the amplitudes of the subprocesses occurring between the two nucleons and of a two-step process in which the excitation of an intermediate eta-meson dominates.