The TAWS system, in addition to an FTLA function for detecting the risk of collision with the terrain, has an end-of-conflict announcement function COT which is activated after the cessation of a warning or alarm concerning the risk of collision with the ground originating from the FTLA function. This COT function, when activated, checks that the aircraft (A) is observing minimum vertical and lateral safe distances, and estimates the lower vertical speed margin with which a new ground collision risk warning will not be retriggered. After confirming the observance of the minimum safe distances, the COT function has an end-of-conflict message (“Clear of terrain”) sent with a lower vertical speed margin indication.