We present a study on the growth and characterization of high-quality single-layer MoS$_2$ with a single orientation, i.e. without the presence of mirror domains. This single orientation of the MoS$_2$ layer is established by means of x-ray photoelectron diffraction. The high quality is evidenced by combining scanning tunneling microscopy with x-ray photoelectron spectroscopy measurements. Spin- and angle-resolved photoemission experiments performed on the sample revealed complete spin-polarization of the valence band states near the K and -K points of the Brillouin zone. These findings open up the possibility to exploit the spin and valley degrees of freedom for encoding and processing information in devices that are based on epitaxially grown materials.