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We report direct evidence of room-temperature ferromagnetic ordering in O-deficient ZnO:Cu films by using soft x-ray magnetic circular dichroism and x-ray absorption. Our measurements have revealed unambiguously two distinct features of Cu atoms associated with (i) magnetically ordered Cu ions present only in the oxygen-deficient samples and (ii) magnetically disordered regular Cu2+ ions present in all the samples. We find that a sufficient amount of both oxygen vacancies (V(O)) and Cu impurities is essential to the observed ferromagnetism, and a non-negligible portion of Cu impurities is uninvolved in the magnetic order. Based on first-principles calculations, we propose a microscopic "indirect double-exchange" model, in which alignments of localized large moments of Cu in the vicinity of the V(O) are mediated by the large-sized vacancy orbitals.