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A systematic study of the Cu-catalyzed chemical vapor deposition of graphene under extremely low partial pressure is carried out. A carbon precursor supply of just PCH4 ~ 0.009 mbar during the deposition favors the formation of large-area uniform monolayer graphene verified by Raman spectra. A diluted HNO3 solution is used to remove Cu before transferring graphene onto SiO2/Si substrates or carbon grids. The graphene can be made suspended over a ~12 μm distance, indicating its good mechanical properties. Electron transport measurements show the graphene sheet resistance of ~0.6 kΩ/□ at zero gate voltage. The mobilities of electrons and holes are ~1800 cm2/Vs at 4.2 K and ~1200 cm2/Vs at room temperature.