Fully-developed magnetohydrodynamic duct flow under a uniform transverse field at high Hartmann numberM is generally well understood, but the case where the duct has two plane, highly-conducting electrodes parallel to the magnetic field, connected by an external conductor of negligible resistance, and two arbitrarily-shaped, insulating walls intersecting the magnetic field shows unusual features. The flow is remarkably sensitive to whether the field line lengthh within the duct varies or not; velocities suddenly change by a factor of orderM as this condition is changed. Ifh varies there are regions of high forwards and backwards flow, whereas ifh is constant the velocities are smaller but can vary in an even more complicated way, with forwards and backwards flow at much higher speeds in the boundary layers on the electrodes. A major novelty in comparison with other high-M duct flows is that the main flow cannot be solved without a scrutiny of these boundary layers. Another novelty is that significant vorticity normal to the imposed field persists. This situation is compared with other duct flows in which such vorticity persists. The stability of the very distorted velocity profiles is not examined.