One-step methane-to-methanol conversion is a "dream reaction"for chemists. In this Letter, continuous direct conversion of methane into methanol with steam under mild conditions using Cu(II)-exchanged mordenite is reported. The methanol production rate depended on the Si/Al ratio, Cu/Al ratio, presence of Brønsted acid sites, and O2 concentration in the feed. Cu-H-mordenite and Cu-Na-mordenite showed comparable methane conversion rates, but the former provided higher methanol selectivity. At a given Si/Al ratio, the methane conversion rate gave a volcano plot as a function of Cu/Al ratio, and methanol selectivity decreased with increasing Cu/Al ratio. These experimental data and DFT calculation implied that binuclear copper species are more active than mononuclear copper species and that copper oxide clusters are favorable for total oxidation.