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We report on a numerical method for the reconstruction of wavefronts in digital holographic microscopy (DHM). It yields simultaneous amplitude and quantitative phase contrast imaging by acquiring a single hologram. The reconstruction method computes the propagation of the complex optical wavefront diffracted by the object and is used to determine the refractive index and/or shape of the object with accuracies in the nanometer range along the optical axis. The technique comprises the recording of a digital hologram of the specimen by means of a standard CCD camera at the exit of a Mach-Zehnder or Michelson type interferometer and an adjustment procedure to determine precisely the reconstruction parameters. The quantitative nature of the reconstructed phase distribution has been demonstrated by an application to surface profilometry where step height differences of a few nanometers have been measured. Another application takes place in biology for transmission phase-contrast imaging of living cells in culture with resolution reaching 30 nanometers in height.