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Single-phase epitaxial Hf_0.5Zr_0.5O₂ films with non-centrosymmetric orthorhombic structure have been grown directly on electrode-free corundum (α-Al₂O₃) substrates by pulsed laser deposition. A combination of high-resolution X-ray diffraction and X-ray absorption spectroscopy confirms the epitaxial growth of high-quality films belonging to the <i>Pca</i>2₁ space group, with [111] out-of-plane orientation. The surface of a 7-nm-thick sample exhibits an atomic step-terrace structure with a corrugation of the order of one atomic layer, as proved by atomic force microscopy. Scanning transmission electron microscopy reveals that it consists of grains with around 10 nm lateral size. The polar nature of this film has been corroborated by pyroelectric measurements. These results shed light on the mechanisms of the epitaxial stabilization of the ferroelectric phase of hafnia.