The problem of pulse pile-up is very often encountered in precise measurements of γ-rays using germanium detectors. The standard method of treating the pile-up events is to identify and reject them using an appropriate electronic system. Digital acquisition techniques now allow the recording of waveforms of pile-up events that can be analyzed and the contributing single pulses recovered, rather than simply tolerating the losses associated with pile-up. In this paper, a method for the off-line digital processing of pile-up events from germanium detectors is demonstrated. The method is based on an appropriate fitting of the detector signals, shaped with a suitable digital pulse shaper. It is shown that the method is able to recover the pile-up events with good accuracy even when the constituent signals are in close proximity. The method is very useful for γ-ray spectroscopy in nuclear physics experiments, where the low intensity signals can be lost due to the pile-up in a high-rate environment.