  The elastic scattering amplitude (ESA) is obtained in the one-loop
approximation of the 2+1 dimensional quantum electrodynamics (QED$_{2+1}$) for
planar charged fermions in an external constant magnetic field. We obtain the
elastic scattering amplitude in the corresponding massive theory and then
discuss and calculate the massless limit of ESA. The imaginary part of ESA are
related to the total probability of photons emission by charged fermions in the
considered magnetic field. Simple analytical formulas are obtained when planar
charged fermions in initial states occupy high excited Landau levels.
