We design and experimentally evaluate a hybrid safe-by-construction collision avoidance controller for autonomous vehicles. The controller combines into a single architecture the respective advantages of an adaptive controller and a discrete safe controller. The adaptive controller relies on model predictive control to achieve optimal efficiency in nominal conditions. The safe controller avoids collision by applying two different policies, for nominal and out-of-nominal conditions, respectively. We present design principles for both the adaptive and the safe controller and show how each one can contribute in the hybrid architecture to improve performance, road occupancy and passenger comfort while preserving safety. The experimental results confirm the feasibility of the approach and the practical relevance of hybrid controllers for safe and efficient driving.