We report the practical evaluation of the compression in a dispersive single-mode fiber of a weak, femtosecond pulse by its interaction with a longer, intense, phase-modulating pulse. Our numerical analysis takes into account the group-velocity dispersion, unavoidable in real fibers, which affects the chirp linearity and hence the extent of compression attainable. We show that, even in the absence of matching of the pulse group velocities, a weak 50-fs pulse of any wavelength between 315 and 800 nm can be compressed from 18.5 to 6.8-fs by a 100-fs, 250-kW pulse at 630 nm. When the group velocities are matched, this compression is superior to that attainable by the less flexible, conventional self-phase modulation technique. In addition, it is shown that a weak, femtosecond near-infrared pulse propagating in the anomalous dispersion region can be compressed by fiber propagation alone, almost without being influenced by group velocity mismatching.