  The fermionic dispersion relation in the presence of a background magnetic
field and a high temperature QED plasma is calculated exactly in the external
field, using the Hard Thermal Loop effective action. As the field strength
increases there is a smooth transition from the weak-field ($qB\ll q^2T^2$)
thermal dispersion relations to the vacuum Landau levels when the background
field is much stronger than any thermal effects ($qB\gg q^2T^2$). The
self-energy at finite field strength acquires an imaginary part. The spectral
width becomes important for critical field strengths ($qB \sim q^2T^2$),
necessitating the use of the full spectral function. It is shown that the
spectral function satisfies the usual condition of normalization and causality.
Using the exact spectral function I also show that the production of chirality
in an external electromagnetic field at high temperature is unaffected by the
presence of the thermal masses of the fermions.
