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We report broken time-reversal symmetry in the pseudogap state of underdoped single crystalline YBa₂Cu₃O_y using muon spin relaxation technique. Slow magnetic fluctuations are observed below the characteristic pseudogap onset temperature T^*, signaled by the Redfield relation between the Lorentzian relaxation rate of the positive muons and the applied field along the muon initial spin polarization. We measured the temperature dependence of the local field and compared it with the order parameter measured by neutron scattering. The slow frequency scale of the fluctuations, 5 orders of magnitude below the energy scale of the pseduogap energy, implies that time-reversal symmetry is broken for properties intermediate between the two scales, but raises the question of the nature of the quantum fluctuations responsible for time-reversal not being absolutely broken.