The path following problem for a car-like robot is considered. The control goal is to bring the robot to a pre-assigned curvilinear path and to stabilize its motion along the path. A new canonical change of variables is suggested. It reduces the problem of stabilizing robot's motion to that of stability of the zero solution of the transformed system in the form that admits feedback linearization. A new control law is synthesized that ensures linearity of the closed-loop system and stabilizes robot's motion along a given target path if the initial conditions belong to a known region. Comparison of the new control law with two earlier obtained linearizing feedbacks known from the literature demonstrates its unquestionable advantages.