The temporal magnetization-direction fluctuations in the three-dimensional classical ferromagnetic Heisenberg model have been generated by Monte Carlo simulation and analyzed by the rescaled-range method to yield the Hurst exponent H. A value of H1 has been found to apply in the ferromagnetic phase characterizing fractional Brownian motion, whereas a value H0.5, reflecting ordinary Brownian motion, applies in the paramagnetic phase. A field-induced crossover from fractional to ordinary Brownian motion has been observed in the ferromagnetic phase.