Although CP violation in the B meson system has been well established by the B factories, there has been no direct observation of time reversal violation. The decays of entangled neutral B mesons into definite flavor states (B0 or B¯0), and J/ψK0S or cc¯K0S final states (referred to as B+ or B−), allow comparisons between the probabilities of four pairs of T-conjugated transitions, for example, B¯0→B− and B−→B¯0, as a function of the time difference between the two B decays. Using 468 million BB¯ pairs produced in Υ(4S) decays collected by the BABAR detector at SLAC, we measure T-violating parameters in the time evolution of neutral B mesons, yielding ΔS+T=−1.37±0.14(stat.)±0.06(syst.) and ΔS−T=1.17±0.18(stat.)±0.11(syst.). These nonzero results represent the first direct observation of T violation through the exchange of initial and final states in transitions that can only be connected by a T-symmetry transformation.