  We present a calculation of the quasiparticle contribution to the
longitudinal thermal conductivities as well as transverse (Hall) thermal
conductivity of an extreme type-II superconductor in a high magnetic field and
at low temperatures. In the limit of frequency and temperature approaching
zero, both longitudinal and transverse conductivities upon entering the
superconducting state undergo a reduction from their respective normal state
values by the factor $(\Gamma /\Delta)^2$, which measures the size of the
region at the Fermi surface containing gapless quasiparticle excitations. We
use our theory to numerically compute the longitudinal transport coefficient in
borocarbide and A-15 superconductors. The agreement with recent experimental
data on LuNi_2B_2C is very good.
