The time-dependent spectrum of neutrons in the water-moderated Oak Ridge Electron Linear Accelerator (ORELA) target is calculated using a modified version of the MORSE multigroup Monte Carlo code with an analytic hydrogen scattering model. Distributions of effective neutron distance traversed in the target are estimated with a time- and energy-dependent algorithm from the leakage normal to the target face. The 10-eV to 20-MeV energy range is adequately represented in the MORSE code by the 174-group VITAMIN-E cross-section library with a P5 expansion. An approximate representation of the ORELA positron source facility, recently installed near the target, is included in the calculations to determine any perturbations the positron source might create in the computed neutron distributions from the target. A series of coupled Monte Carlo calculations is performed from the target to the positron source and back to the target using a next-event estimation surface source for each step. The principal effect of the positron source is an increase in the distance for the lower energy neutron spectra, producing no real change in the distributions where the ORELA source is utilized for experiments. Different configurations for the target are investigated to simulate the placement of a shadow bar in the neutron beam. These beam configurations include neutrons escaping from (a) the central tantalum plates only, (b) the entire target with the tantalum plates blocked out, and (c) only a small area from the water. Comparisons of the current data with previous calculations having a less detailed model of the tantalum plates are satisfactory.