  The thermal self-energy of an electron in a static uniform magnetic field $B$
is calculated to first order in the fine structure constant $\alpha $ and to
all orders in $eB$. We use two methods, one based on the Furry picture and
another based on Schwinger's proper-time method. As external states we consider
relativistic Landau levels with special emphasis on the lowest Landau level. In
the high-temperature limit we derive self-consistent dispersion relations for
particle and hole excitations, showing the chiral asymmetry caused by the
external field. For weak fields, earlier results on the ground- state energy
and the anomalous magnetic moment are discussed and compared with the present
analysis. In the strong-field limit the appearance of a field-independent
imaginary part of the self-energy, related to Landau damping in the
$e^{+}e^{-}$ plasma, is pointed out.
