Method for the production of a multi-layer metal cord that is rubberised in situ using an unsaturated thermoplastic elastomer

Method of manufacturing a multi-layer metal cord having a plurality of concentric layers of wires, comprising one or more inner layer(s) and an outer layer, of the type "rubberized in situ. The method includes the following steps: at least one step of sheathing at least one inner layer with the rubber or the rubber composition by passing through at least one extrusion head; and an assembling step in which the wires of the outer layer are assembled around the inner layer adjacent to it, in order to form the multi-layer cord thus rubberized from the inside. The rubber is an unsaturated thermoplastic elastomer extruded in the molten state, preferably a thermoplastic elastomer of the thermoplastic stirene (TPS) elastomer type such as an SBS, SBBS, SIS or SBIS block copolymer for example.