- We consider the stability of axially unbounded cylindrical flows that contain a toroidal magnetic background field with the same radial profile as their azimuthal velocity. For ideal fluids, Chandrasekhar had shown the stability of this configuration if the Alfven velocity of the field equals the velocity of the background flow, i.e., if the magnetic Mach number Mm = 1. We demonstrate that magnetized Taylor-Couette flows with such profiles become unstable against non-axisymmetric perturbations if at least one of the diffusivities is finite. We also find that for small magnetic Prandtl numbers Pm the lines of marginal instability scale with the Reynolds number and the Hartmann number. In the limit Pm → 0 the lines of marginal instability completely lie below the line for Mm = 1 and for Pm → ∞ they completely lie above this line. For any finite value of Pm, however, the lines of marginal instability cross the line Mm = 1, which separates slow from fast rotation. The minimum values of the field strength and the rotation rate that are needed for the instability (slightly) grow if the rotation law becomes flat. In this case, the electric current of the background field becomes so strong that the current-driven Tayler instability (which also exists without rotation) appears in the bifurcation map at low Hartmann numbers. © 2015. The American Astronomical Society. All rights reserved.