This paper presents a new obstacle-avoidance method for mobile robots. This approach, called the FEB (Fast Elastic Band) method, has been developed and successfully tested on the experimental mobile robot PHOPE-1S. The FEB method eliminates the shortcomings of the elastic band method previously introduced, yet retains all the advantages of its predecessor. The FEB algorithm is computationally efficient, and it allows continuous and fast motion of the mobile robot without stopping for obstacles. The FEB-controlled mobile robot traverses very densely cluttered obstacle courses and is able to pass through narrow openings or narrow corridors without oscillations. The results of the simulation and experiment have verified the validity of the proposed method.