Epitaxial growth of rhombohedral boron nitride (r-BN) on different polytypes of silicon carbide (SiC) is demonstrated using thermally activated hot-wall chemical vapour deposition and triethyl boron and ammonia as precursors. With respect to the crystalline quality of the r-BN films, we investigate the influence of the deposition temperature, the precursor ratio (N/B) and the addition of a minute amount of silicon to the gas mixture. From X-ray diffraction and transmission electron microscopy, we find that the optimal growth temperature for epitaxial r-BN on the Si-face of the SiC substrates is 1500 degrees C at a N/B ratio of 642 and silicon needs to be present not only in the gas mixture during deposition but also on the substrate surface. Such conditions result in the growth of films with a c-axis identical to that of the bulk material and a thickness of 200 nm, which is promising for the development of BN films for electronic applications.