Hexagonal boron nitride (h-BN) has a potential for optical device applications in the deep ultraviolet spectral region. For several decades, only amorphous and turbostratic boron nitride (BN) films had been grown by chemical vapor deposition and metalorganic vapor phase epitaxy. By introducing flow-rate modulation epitaxy (FME), which enables us to reduce parasitic reactions and lower the optimal growth temperature, we have succeeded in growing single-phase h-BN epitaxial films on nearly lattice-matched (111) Ni substrates. The h-BN epitaxial films exhibit near-band-gap ultraviolet luminescence at a wavelength of 227nm in cathodoluminescence at room temperature. The combination of FME and the lattice-matched substrate paves the way for the epitaxial growth of high-quality h-BN.