A sealing device for providing the upstream/downstream sealing of a valve comprising a valving member movable inside the bore of a tubular body and comprising at its periphery a sealing surface coming to bear on a sealing packing mounted in the body at the end of the valving stroke. This sealing packing comprises a dynamic annular sealing element and a coronal metal membrane, whose outer edge portion has flat radial part applied axially against a steep side of a shoulder in the bore of the body, for providing static sealing therewith.