Chemical vapor deposition (CVD) synthesis of large-domain hexagonal boron nitride (h-BN) with a uniform thickness is very challenging, mainly due to the extremely high nucleation density of this material. Herein, we report the successful growth of wafer-scale, high-quality h-BN monolayer films that have large single-crystalline domain sizes, up to ~72 μm in edge length, prepared using a folded Cu-foil enclosure. The highly confined growth space and the smooth Cu surface inside the enclosure effectively reduced the precursor feeding rate together and induced a drastic decrease in the nucleation density. The orientation of the as-grown h-BN monolayer was found to be strongly correlated to the crystallographic orientation of the Cu substrate: the Cu (111) face being the best substrate for growing aligned h-BN domains and even single-crystalline monolayers. This is consistent with our density functional theory calculations. The present study offers a practical pathway for growing high-quality h-BN films by deepening our fundamental understanding of the process of their growth by CVD.