Current Full Range Adaptive Cruise Control (FRACC) systems switch between separate adaptive cruise control and collision avoidance systems. This can lead to jerky responses and discomfort during the transition between the two control modes. We propose a Full Range Adaptive Cruise Control (FRACC) design integrating adaptive cruise control and collision avoidance into a single non-linear mathematical formulation. The proposed FRACC responds to a velocityerror using a sigmoidal function of forward spacing. Mathematical properties of the controller, in particular string stability, are examined. Simulation experiments demonstrate that the controller yields smooth and safe responses in typical highway scenarios, including hard-braking and cut-in scenarios. Results also show a clear advantage of the proposed controller in string stability performance with reference to a state-of-The-Art controller.