The 13 TW pulsed electron beam generated by Hermes III [J. J. Ramirezetal.,DigestofTechnicalPapers, 6th IEEE Pulsed Power Conference (IEEE, New York, 1987), pp. 294] is transported 10.8 m in a low‐pressure gas with 79±1.5±[5]% energy transport efficiency. The uncertainties refer to the rms shot‐to‐shot variation and estimated systematic error, respectively. The high efficiency obtained is accomplished by removing the inward momentum of the beam at injection via an active magnetic lens. The reduced transverse momentum permits self‐magnetic field confinement at low pressures with low inductive and collisional loss. The region of efficient transport lies between regions of instability. Consistent with experiment, the analytic and numerical models developed here predict that them=1 resistive hose instability degrades transport above 100 Torr and plasma return‐current instabilities disrupt the beam below 1 Torr. Within this pressure ‘‘window’’ of stable propagation, the models explain the mechanisms responsible for maximum transport.