We solve an open problem of output-feedback stabilization for a nonlinear system that is not uniformly observable at the target equilibrium. The plant's model is of interest not only because it is an instance of the well-studied class of bilinear non-uniformly observable systems, but also because it represents the dynamics of an antiblock-braking system (ABS). To stabilize the origin, where the system is unobservable, we use a switched observer previously reported and a novel hybrid controller. The former relies on making the system operate in a regime that induces enough excitation to ensure the state estimation, while the latter is based on tracking a vanishing piecewise-constant reference. Our main statement establishes asymptotic stability of the origin in a hybrid sense.