The invention relates to motor vehicles which are equipped with an electric power steering system. Using a known method, the value of the maximum current that is allowed to travel through the electric assist motor is calculated at each instant (t), and a threshold value is determined for said maximum calculated current. Once the value of the maximum calculated current at each instant reaches the threshold value, a gradually-decreasing percentage is applied to the assist current calculated as a function of the torque (C) applied to the steering wheel. In this way, the assist current is reduced before the assist motor thermal protection procedure is activated, such that the driver is not surprised by a sudden lack of power assistance.