An equation describing the evolution of the transverse energy of a segment of a paraxial axisymmetric relativistic electron beam (REB) propagating in a gas-plasma scattering medium along an external magnetic field is used to find the equation relating the final equilibrium radius of the beam to its initial nonequilibrium value. An analytical expression for the increase in the mean-square emittance of an REB during transport up to achievement of the equilibrium state is found for the case considered. The dependence of the final equilibrium radius and the corresponding increase in the mean-square emittance on the density of the scattering medium and the induction of the external magnetic field is investigated.