We study the self-modulation of a circularly polarized Alfven wave in a strongly magnetized relativistic electron-positron plasma with finite temperature. This nonlinear wave corresponds to an exact solution of the equations, with a dispersion relation that has two branches. For a large magnetic field, the Alfven branch has two different zones, which we call the normal dispersion zone (where d omega/dk > 0) and the anomalous dispersion zone (where d omega/dk < 0). A nonlinear Schrodinger equation is derived in the normal dispersion zone of the Alfven wave, where the wave envelope can evolve as a periodic wave train or as a solitary wave, depending on the initial condition. The maximum growth rate of the modulational instability decreases as the temperature is increased. We also study the Alfven wave propagation in the anomalous dispersion zone, where a nonlinear wave equation is obtained. However, in this zone the wave envelope can evolve only as a periodic wave train.