We have developed a digital holographic microscope (DHM), in a transmission mode, adapted to the quantitative study of cellular dynamics. Living cells are optically probed by measuring the phase shift they produce on the transmitted wave front. The high temporal stability of the phase signal, equivalent to lambda/1800, and the low acquisition time (down to 20 mu s) enables to monitor cellular dynamics processes. An experimental procedure allowing to calculate both the intracellular refractive index and the cellular thickness (morphometry) from the measured phase shift is presented.