In real single-mode optical fibers, imperfections cause the two possible polarizations to propagate at different phase velocities. This birefringence leads to different group velocities. We have measured the resulting mode dispersion in short fiber lengths (0.5-2.5 m) from the depolarization of broad-bandwidth light. In a typical fiber we found 30 psec/km at 0.69-microm wavelength, in good agreement with the observed birefringence. The effect of mode dispersion can be compensated by a +/-68 degrees double twist midway along the fiber, interchanging the fast and slow modes.