The observations of CP-symmetry breaking, first in neutral K decays and more recently in B mesons, are consistent with the standard model. Local Lorentz invariant quantum field theories imply CPT invariance, in accordance with all experimental evidence. Hence, it is expected that the CP-violating weak interaction also violates time-reversal invariance. Experiments that could provide direct evidence supporting T noninvariance, without using an observation which also violates CP, involve either nonvanishing expectation values of T-odd observables (upper limits for electric dipole moments of the neutron and the electron), or the exchange of initial and final states, which are not CP conjugates to each other, in the time evolution for transition processes. The latter, requiring neutrinos or unstable particles, are particularly difficult to implement. We report the direct observation of T violation in the B meson system, through the exchange of initial and final states in transitions that can only be connected by a T-symmetry transformation. For this study we used 468 10^6 BBbar pairs produced in Y(4S) decays collected by the BABAR detector at the PEP-II asymmetric-energy e+e- colliderSLAC. Using the entanglement of the B mesons produced in the Y(4S) decay, the "flavor tagging" has been combined, for the first time, with the "CP tagging", for the construction of T-transformed processes.