We report on the performance of a fine-grained 13-ton compensating lead/scintillating-fiber calorimeter, and in particular on its capability of localizing the particles that produce showers in it. The RMS position resolution was found to be 1.7 mm for electromagnetic showers and 5.1 mm for hadronic showers at 80 GeV, averaged over a tower with an effective radius of 39 mm. Pion-pion separation through analysis of the energy deposit pattern was achieved in more than 95% of the cases for distances down to 8 cm at 80 GeV. Because of the good lateral position resolution, detailed information on the longitudinal shower development could be obtained, with the help of tracking information, when single particles entered the (longitudinally unsegmented) detector at a small angle with respect to the fiber axis. This information made it possible to eliminate the effects of light attenuation in the fibers on the hadronic energy resolution and allowed e/π separation at the 10-4 level.