  We have measured the thermal conductivity of the geometrically frustrated
quasi-onedimensional spin system Cu$_3$Mo$_2$O$_9$ in magnetic fields. A
contribution of the thermal conductivity due to spins has been observed in the
thermal conductivity along the spin chains. The thermal conductivity due to
phonons, $\kappa_{\rm phonon}$, has been found to decrease by the application
of a magnetic field, which has been explained as being due to the reduction in
the spin gap originating from the spin-singlet dimers. Moreover, it has been
found that $\kappa_{\rm phonon}$ increases with increasing field in high fields
above ~7 T at low temperatures. This suggests the existence of a novel
field-induced spin state and is discussed in terms of the possible
spin-chirality ordering in a frustrated Mott insulator.
