The device includes a closing element (43) for closing the valve (43, 44), and which is associated with a first float (32, 47) which is raised to open the valve (43, 44); and a second float (58) concentric with the first float (32, 47), and which is raised to fully discharge the tank (5). The first float (32, 47) is defined by the overflow discharge tube (32), the bottom end (34) of which is normally closed by a disk (47) by virtue of the action of a spring (51). The second float (58) is normally locked upwards by a hook element (67) engaging a fixed stop (66), so as not to affect the downward return movement of the first float (32, 47), and so partially discharge the tank (5). Conversely, when the hook element (67) is released, the second float (58) draws the discharge tube (32) upwards, and delays closing of the discharge tube to fully discharge the tank (5).