We constructed an efficient source of photon pairs using a waveguide-type nonlinear device and performed a two-photon interference experiment with an unbalanced Michelson interferometer. As the interferometer has two arms of different lengths, photons from the short arm arrive at the detector earlier than those from the long arm. We find that the arrival time difference (Delta L/c) and the time window of the coincidence counter (Delta T) are important parameters which determine the boundary between the classical and quantum regimes. Fringes of high visibility ( 80% +/- 10%) were observed when Delta T < Delta L/c. This result is explained only by quantum theory and is clear evidence for quantum entanglement of the interferometer's optical paths.