We have investigated the superconducting state of the noncentrosymmetric compound Re6Zr using magnetization, heat capacity, and muon-spin relaxation or rotation (μSR) measurements. Re6Zr has a superconducting transition temperature, Tc=6.75±0.05 K. Transverse-field μSR experiments, used to probe the superfluid density, suggest an s-wave character for the superconducting gap. However, zero and longitudinal-field μSR data reveal the presence of spontaneous static magnetic fields below Tc indicating that time-reversal symmetry is broken in the superconducting state and an unconventional pairing mechanism. An analysis of the pairing symmetries identifies the ground states compatible with time-reversal symmetry breaking.