This invention provides an apparatus for the control of acceleration and deceleration of an automobile, which enlarges the ratio of increase in the real speed of the automobile while the automobile is accelerated from its auto-cruise condition and precludes overshoot and hunting of the real speed of the automobile after the acceleration or deceleration is completed and the resumption of constant-speed travel is started. In the course of acceleration, the throttle opening angle is first increased abruptly by a predetermined amount and then increased as the function of time and, upon disappearance of an acceleration signal, the throttle opening angle is decreased by an amount based on the real speed of the automobile as it exists at that moment. In the course of deceleration, the throttle opening angle is decreased abruptly to a very small value which may be almost zero and, upon disappearance of a deceleration signal, the throttle opening angle is increased to a level based on the real speed of the automobile as it exists at that moment.