The vehicle-use speed control apparatus sets a virtual preceding vehicle assumed to run at a target speed at a predetermined initial distance ahead of the own vehicle when the brake pedal is operated causing a relative speed between the actual vehicle speed and the target speed exceeds a predetermined value, calculates an initial value of a performance index for approach and alienation based on the initial distance and the target speed. Thereafter, the vehicle-use speed control apparatus repeatedly calculates a following distance to the virtual preceding vehicle based on a time elapsed from when a speed control start condition is satisfied, the current relative speed and the initial distance, and a target relative speed based on the initial distance, the initial value of the performance index and the following distance, and controls the vehicle speed based on the difference between the target relative speed and the actual relative speed.