A method for the co-electrolysis of steam H 2 O and of carbon dioxide CO 2, and where appropriate in-situ methanation, implemented in a reactor comprising a stack of individual electrolysis cells (C1, C2) of solid oxide type, with a rectangular or square area, each formed of a cathode (2.1, 2.2) comprising material(s) for catalyzing a methanation reaction, of an anode (4.1, 4.2) and of an electrolyte (3.1, 3.2) inserted between the cathode and the anode, a plurality of electrical and fluid interconnectors each arranged between two adjacent individual cells (C1, C2) with one face thereof in electrical contact with the anode (4.1) of one of the two individual cells and the other face thereof in electrical contact with the cathode (2.2) of the other of the two individual cells, and a plurality of electrical contact and gas distribution elements, each arranged between a cathode and an interconnector, according to which: - a first zone (61, 71, 81) and a second zone (62, 72, 82) of each interconnector are supplied independently (EH2; EH2), with a mixture of steam H 2 O and of carbon dioxide CO 2 and it is distributed to the cathode of each individual cell, then the synthesis gas produced (mixture of carbon monoxide CO and hydrogen H 2) and where appropriate the additional mixture of methane CH 4 and of steam H 2 O produced by methanation are recovered (SH2; SH2) within the cathode itself, in a third zone (63, 73, 83) and a fourth zone (64, 74, 84) of each interconnector in fluid communication respectively with the first zone and the second zone; each electrical contact and gas distribution element integrating a sealing bead forming a gas distribution barrier separating a first gas flow sector comprising the first and third zones from a second gas flow sector comprising the second and fourth zones, the first and second sectors being adjoined by the barrier, forming an area substantially equal to that of each cell; the first to fourth zones being dimensioned, and the gas circulation barrier being arranged, such that the flow section of the gases increases or decreases between the first and third zones within the first sector and the flow section of the gases also increases or decreases between the second and fourth zones within the second sector; - the first and second zones of each interconnector are supplied such that the circulation to each cathode in the first flow sector is in counterflow to the circulation in the second flow sector. - a first zone (61, 71, 81) and a second zone (62, 72, 82) of each interconnector are supplied independently (EH2; EH2), with a mixture of steam H 2 O and of carbon dioxide CO 2 and it is distributed to the cathode of each individual cell, then the synthesis gas produced (mixture of carbon monoxide CO and hydrogen H 2) and where appropriate the additional mixture of methane CH 4 and of steam H 2 O produced by methanation are recovered (SH2; SH2) within the cathode itself, in a third zone (63, 73, 83) and a fourth zone (64, 74, 84) of each interconnector in fluid communication respectively with the first zone and the second zone; each electrical contact and gas distribution element integrating a sealing bead forming a gas distribution barrier separating a first gas flow sector comprising the first and third zones from a second gas flow sector comprising the second and fourth zones, the first and second sectors being adjoined by the barrier, forming an area substantially equal to that of each cell; the first to fourth zones being dimensioned, and the gas circulation barrier being arranged, such that the flow section of the gases increases or decreases between the first and third zones within the first sector and the flow section of the gases also increases or decreases between the second and fourth zones within the second sector; - the first and second zones of each interconnector are supplied such that the circulation to each cathode in the first flow sector is in counterflow to the circulation in the second flow sector.