Collisions of cold and ultracold BH in the v= 0 level with the He atom are investigated using the quantum mechanical scattering formulation. The elastic and the inelastic cross sections are calculated using the two-dimensional ab initio potential energy surface. It is shown that the elastic cross section is larger than the inelastic one. When the collision energy is very low, the elastic cross section follows the Wigner threshold law and is one order of magnitude larger than that of He-O2, while it is much smaller than that of He--H2. The efficiency of the rotationally quenching state is given. The △j = -1 transition is most efficient. The resonances are also found to occur at about the same translational energy （0.1-1 cm-1）, which gives rise to steps in the rate coefficient at temperatures around 0.1-1 K.
