A coherent state of the Dirac oscillator in (1 + 1) dimensions is constructed. This is a relativistic extension of Schrödinger's coherent state of the nonrelativistic harmonic oscillator. There are interesting differences between the Dirac and the Schrödinger cases. In the Dirac case, the coherent state spreads out in time but after a while the original coherent state is nearly restored. Then it spreads again. This process recurs but in an aperiodic manner. In addition, the expectation value of coordinate x for the coherent state exhibits Zitterbewegung. The possibility of an analogue of Ehrenfest's theorem is discussed.