We study so-called "hybrid feedback stabilizers" for an arbitrarily general system of linear differential equations. We prove that under assumptions of controllability and observability there exists a hybrid feedback output control which makes the system asymptotically stable. The control is designed by making use of a discrete automaton implanted into the systems dynamics. In general, the automaton has infinitely many locations, but it gives rise to a "uniform" (in some sense) feedback control. The approach we propose goes back to classical feedback control techniques combined with some ideas used in stability theory for equations with time-delay.