We performed Brillouin light scattering spectroscopy on a single crystal of BiFeO_3 (BFO) in the temperature range from 110 K to 300 K and observed the Brillouin scattering from the longitudinal acoustic (LA) and transverse acoustic (TA) phonon modes, which were traveling along the c-axis. We determined the elastic constants for the LA mode (C_33) and TA mode (C_44) to be approximately 251 GPa and 90.3~103 GPa at 300 K, respectively. We observed splitting of the degenerate TA modes (corresponding to the elastic constant C_44), which is not expected for the TA modes traveling along the three-fold symmetry axis (the c-axis) of BFO. We also present the temperature dependences of the frequencies of the LA and TA modes, which include anomalies such as the LA mode softening and TA mode frequency drop at around 210 K.