  We study the stability of cylindrical Taylor-Couette flow in the presence of
azimuthal magnetic fields, and show that one obtains non-axisymmetric
magnetorotational instabilities, having azimuthal wavenumber m=1. For
Omega_o/Omega_i only slightly greater than the Rayleigh value (r_i/r_o)^2, the
critical Reynolds and Hartmann numbers are Re_c ~ 10^3 and Ha_c ~ 10^2,
independent of the magnetic Prandtl number Pm. These values are sufficiently
small that it should be possible to obtain these instabilities in the PROMISE
experimental facility.
