Electromagnetic calorimeter modules based on a uniform array of plastic scintillating fibers embedded in a lead alloy have been built and tested. Techniques have been developed to assemble large volumes of this composite material and to machine it into the tapered trapezoidal modules appropriate for modern, hermetically-sealed calorimeters with pointing geometry. Using this technique, a 300-element array of such modules has been built and instrumented. Prototypes and subsets of the larger array have been tested in electron and photon beams ranging in energy from 0.035 to 5.0 GeV. Improvements in fiber characteristics, assembly procedures, and geometrical optimization have led to substantial performance gains over previous similar detectors. The average resolution of these detectors is determined to be σ/ E ≈ 6.3%/√ E (GeV). The techniques of fabrication and the detector tests are described.