The method comprises the steps of: impregnating the part with a liquid impregnation composition containing at least a phosphate type compound, at least via a fraction of the outside surface of the part; applying a coating composition on said fraction of the outside surface of the part, the coating composition comprising a colloidal suspension of at least one refractory oxide in water, at least one compound essentially of the borosilicate type in powder form and having healing properties, and at least one metallic boride in powder form; and applying heat treatment after applying the coating composition.