A nucleonic analysis of the Engineering Test Facility vacuum pumping (VP) duct shield has been performed using a coupled Monte Carlo/discrete ordinates method. This method used Monte Carlo to determine internal and external boundary surface sources for subsequent use in discrete ordinates calculations of the neutron and gamma-ray transport through the shield. Confidence in both the coupled method and the results obtained through its use was provided through comparisons with three-dimensional Monte Carlo results. Calculations in the toroidal plasma chamber and the neutral beam injector duct, which were required prior to the VP duct shield calculations, are also described.