A fluid dispenser device with a body (1), and an axially movable dispenser head (2). The body receives a reservoir (10) having a tube (11) with a proximal axial opening (12) closed by a stopper (20) and adapted to slide in leaktight manner in the tube (11). The dispenser head includes a cannula (4) connected at one end to the dispenser orifice (3), and provided at its other end with a perforating tip (5) for perforating the stopper. The cannula is inserted into a cannula support (9) secured to the dispenser head and including a borehole with an inside diameter equal to, or a little greater than, the outside diameter of the cannula. The distal end of the cannula support (9) has a projection extending inwards and an inside diameter equal to, or a little less than, the outside diameter of the cannula.
