We experimentally observe a spontaneous temporal symmetry breaking instability in a coherently-driven passive optical Kerr resonator. The cavity is synchronously pumped by time-symmetric pulses yet we report output pulses with strongly asymmetric temporal and spectral intensity profiles, with up to 71% of the energy on the same side of the pump center frequency. The instability occurs above a certain pump power threshold but remarkably vanishes above a second threshold, in excellent agreement with theory. We also observe a generalized bistability in which an asymmetric output state coexists with a symmetric one in the same pumping conditions.