  Simulations and analytic arguments suggest that the turbulence driven by
magnetorotational instability (MRI) in accretion discs can amplify the toroidal
(azimuthal) component of the magnetic field to a point at which magnetic
pressure exceeds the combined gas + radiation pressure in the disc. Arguing
from the recent analysis by Pessah and Psaltis, and other MRI results in the
literature, we conjecture that the limiting field strength for a thin disc is
such that the Alfven speed roughly equals the geometric mean of the Keplerian
speed and the gas sound speed. We examine the properties of such
magnetically-dominated discs, and show that they resolve a number of
outstanding problems in accretion disc theory. The discs would be thicker than
standard (Shakura-Sunyaev) discs at the same radius and accretion rate, and
would tend to have higher colour temperatures. If they transport angular
momentum according to an alpha-prescription, they would be stable against the
thermal and viscous instabilities that are found in standard disc models. In
discs fuelling active galactic nuclei, magnetic pressure support could also
alleviate the restriction on accretion rate imposed by disc self-gravity.
