Exciton spin relaxations in bulk GaN were directly observed with sub-picosecond's time resolution. The obtained spin relaxation times of A-band free exciton are 0.47 ps - 0.25 ps at 150 K - 225 K. The spin relaxation time of the acceptor bound exciton at 15K is measured to be 1.1 ps. These are at least one order of magnitude shorter than those of the other III-V compound semiconductors. The spin relaxation time of A-band free exciton is found to be proportional to T1.4, where T is the temperature. The fact that the spin relaxation time in GaN is shorter than that in GaAs, in spite of the small spin-orbit splitting, suggests that the spin relaxation is dominated by the defect-assisted Elliot-Yafet process.