Gravitational lensing of distant sources by massive compact objects such as stars produces multiple images which are too closely spaced to resolve. The combined image can be magnified however, resulting in an increase in apparent luminosity. This has been thought to be the sole signature for such "microlensing" events. We propose here a new signature which may be more sensitive and could give an unambiguous determination of the mass of the lensing object, independent of its distance. A time delay between the superposed images on the order of microseconds to milliseconds will occur for lensing by objects in our galaxy. This might be measurable in sources such as radio pulsars, some stars, and perhaps even supernovae. We calculate in some detail microlensing probabilities (appropriate for any study of microlensing) from objects in the Galactic disk and bulge, and also halo objects such as massive stars or large black holes which may contribute significantly to the dark matter in our galaxy. We suggest that a study of such time delays could resolve many features of the mass distribution of luminous and dark matter in our galaxy and might even provide new tests of general relativity.