The invention relates to a single-unit faucet that comprises a tubular faucet body (2) and a thermostatic cartridge (3) defining a pre-assembled assembly provided inside the faucet body and including a hollow shell (4) made of a plastic material, in which is provided a thermostatic means (5) for adjusting the temperature of a cold-fluid and hot-fluid mixture. In order to firmly and simply secure the shell in the faucet body, a rigid part (6) is provided between them, and is capable of movement along a movement axis (Z-Z) transverse to the longitudinal axis (X-X) of the faucet body, while being kinematically connected to the shell both in the direction of the axis of the faucet body and in rotation about said axis. The rigid part includes a bore (63) centred on the movement axis and opening on a face (62B) of the rigid part oriented towards the faucet body, in order to interact by screwing with the threaded rod (71) of a screw (7) for moving the rigid part until a bearing portion of said face is pressed against the lower face (2A) of the faucet body when the screw is inserted from outside the faucet body into a transverse through hole (21) of the faucet body, which is smaller than the head (72) of the screw.