In superconducting materials, electrical resistivity vanishes abruptly at a critical temperature, T_c. Below this temperature, electrons in the ground state allow current to flow without dissipation, regardless of any thermal excitations that always exist at finite temperature. In the recently suggested superinsulating state, a true zero-current state can never exist due to residual conduction mechanisms. If, however, these residual conduction mechanisms are weak enough, it may be possible to observe the signature of the transition to this intriguing state. We report on the observation of a sharp drop in conductance of several orders of magnitude, occurring at T<0.05K, in a disordered superconductor in the vicinity of the magnetic-field induced superconductor-insulator transition. This phenomenon is observed at temperatures and magnetic fields where electron-phonon coupling is strongly suppressed.