A following distance control device includes a vehicle speed detecting device that detects the vehicle speed VSPD of an own vehicle and outputs a vehicle speed equal to or less than a predetermined speed as zero, a following distance detecting device that detects a following distance between a preceding vehicle and the own vehicle, a first requested acceleration computing device that computes a requested acceleration GBASE from the vehicle speed and the following distance, and a stop determining device that determines the stop of the own vehicle, on the basis of a requested acceleration Greq_pre_0 kph at a time point when the output value of the vehicle speed detecting device becomes zero.