A device executes a preceding vehicle following driving control and successively calculates a friction-braking-start threshold value, an engine-braking-start threshold value, an acceleration-control-completion threshold value by using equations L1, L2 and L3, respectively. The device uses a constant vehicle speed zone determined between the engine-braking-start threshold value and the acceleration-control-completion threshold value. When a compensated-evaluation-index value KdB_c of a vehicle distance between the own vehicle and a preceding vehicle is within the constant vehicle speed zone, the device drives the own vehicle at a constant speed. This control suppresses fuel consumption because of decreasing an acceleration and deceleration frequency of the own vehicle. Decreasing the deceleration frequency makes it possible to provide good vehicle following because less frequent acceleration is needed for following the preceding vehicle even if another vehicle cuts in between the own vehicle and the preceding vehicle or the preceding vehicle accelerates after decelerating.