  The very-low-temperature thermal conductivity (\kappa) is studied for
BaCo_2V_2O_8, a quasi-one-dimensional Ising-like antiferromagnet exhibiting an
unusual magnetic-field-induced order-to-disorder transition. The nearly
isotropic transport in the longitudinal field indicates that the magnetic
excitations scatter phonons rather than conduct heat. The field dependence of
\kappa shows a sudden drop at \sim 4 T, where the system unndergoes the
transition from the N\'eel order to the incommensurate state. Another dip at
lower field of \sim 3 T indicates an unknown magnetic transition, which is
likely due to the spin-flop transition. Moreover, the \kappa(H) in the
transverse field shows a very deep valley-like feature, which moves slightly to
higher field and becomes sharper upon lowering the temperature. This indicates
a magnetic transition induced by the transverse field, which however is not
predicted by the present theories for this low-dimensional spin system.
