  The influence of a toroidal magnetic field on the dynamics of Rossby waves in
a thin layer of ideal conductive fluid on a rotating sphere is studied in the
"shallow water" magnetohydrodynamic approximation for the first time.
Dispersion relations for magnetic Rossby waves are derived analytically in
Cartesian and spherical coordinates. It is shown that the magnetic field causes
the splitting of low order (long wavelength) Rossby waves into two different
modes, here denoted fast and slow {\em magnetic Rossby waves}. The high
frequency mode (the fast magnetic Rossby mode) corresponds to an ordinary
hydrodynamic Rossby wave slightly modified by the magnetic field, while the low
frequency mode (the slow magnetic Rossby mode) has new and interesting
properties since its frequency is significantly smaller than that of the same
harmonics of pure Rossby and Alfv{\'e}n waves.
