A nuclear reactor includes a plurality of mechanisms (11) that drive the contact members (9) that control the reactivity, of the core. Each mechanism includes a driving member (21) including a driving part (23) forming one out of a screw or a nut, a member (27) for applying a rotary torque of the rotor (19) to the driving member (21), a driven member (29) translationally connected to one of the control members (9) and including the other out of screw and a nut; and a member (33) that is selectively mobile between a position of blocking the driving member (21) and a position of releasing the driving member (21). In each drive mechanism (11), the motor (15) is fully immersed in the primary coolant inside the vessel (3); the rotor (19) has a central passage (35), the member for applying the rotary torque (27) being situated in or near the central passage (35); the driving member (21) includes a connecting part (37) engaged in the central passage (35) and collaborating with the member for applying the rotary torque (27), the connecting part (37) being free to effect a translational movement inside the central passage (35) with respect to the rotor (19) when the or each blocking member (33) is in the releasing position.