We report direct experimental evidence that the insulating phase of a disordered, yet strongly interacting two-dimensional electron system becomes unstable at low temperatures. As the temperature decreases, a transition from insulating to metal-like transport behavior is observed, which persists even when the resistivity of the system greatly exceeds the quantum of resistivity h/e2. The results have been achieved by measuring transport on a mesoscopic length scale while systematically varying the strength of disorder.