Abstract: The dismounting and/or cutting of each of the components (6) of the internal equipment fastened in the vessel (4) of the reactor are carried out underwater within the vessel (4). Metal elements (12) separated from the vessel (4) are obtained and are successively extracted from the vessel and then compacted underwater in a compacting press. The metal elements (36) obtained after the compacting are deposited underwater in a storage and transport container (37). The dismantling apparatus comprises in particular a handling and cutting assembly (11) which makes it possible to carry out the cutting of cylindrical portions of the core shroud (6) of the reactor and the transport of the portions (12) into the compacting press (34).