The stability of hydromagnetic flow produced in a thin annular region between concentric cylinders by the interaction of a superimposed radial current and an axial magnetic field is examined both when the bounding cylinders are stationary and when rotating. After deriving the small gap approximation equations for the steady‐state one‐dimensional azimuthal flow and for the corresponding induced electric potential, the equations governing the growth of infinitesimal axisymmetric disturbances are obtained and solved at neutral stability by use of the Galerkin method. Stability results for the nonrotating cylinder case are shown to be expressible in terms of existing results for the stability of hydromagnetic flow produced by an azimuthal pressure gradient. It is found, in our thin film approximation, that the radial current must generally exceed a certain critical value before the one‐dimensional flow breaks down into a three‐dimensional pattern. Rotation of the outer cylinder enhances stability while rotation of the inner cylinder produces a greater tendency toward instability as reflected by lower Dean numbers at the onset of secondary flow.