The molecular beam epitaxial growth of a film of the layered semiconductor GaSe onto a near perfect, hydrogenated, 1 x 1 ordered Si(111) substrate has been monitored at its initial steps by reflection high-energy electron diffraction at grazing incidence (RHEED) and checked in situ by X-ray photoemission spectroscopy (XPS). The changes in surface lattice parameter and in core level peak areas as a function of the thickness of the deposited film, within the first two layers, are explained as follows: (i) the single atomic planes of Ga (bound to Si) and Se (above Ga), which substitute H to passivate the Si(111) surface are exact monolayers and keep the same perfect 1 x 1 structure of the Si substrate, (ii) the epitaxial growth of GaSe is strictly a layer-by-layer process and (iii) the lattice mismatch between the passivated substrate and the crystalline GaSe film is fully absorbed within the first elementary "Ga2Se2" layer through an original relaxation process.