A possible smallest single-walled BN nanotube with the diameter of 0.4nm is proposed. The studied nanotube is of armchair (3,3) and thought to be associated with the BN fullerene, B12N12. We have applied the ab initio MP2/6-31G//HF/6-31G method to investigate the growth mechanisms of this nanotube from B12N12. Our results show that a B12N12 fullerene may react with B2H6 and NH3 to form an armchair (3,3) tubular structure under certain experimental conditions such as high temperature. The armchair (3,3) tube can continuously incorporate B and N atoms to grow and can also disassociate H2 to close into a half-B12N12 cap. The growth of the tube is predicted to be favorable to the capping of the tube in an atmosphere with B2H6 and NH3 in it. And the cap of the tube is relatively easier to be opened by H2 or B2H6 and NH3 to convert to a growing open-ended tube.