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 (Γ /Δ)^2, which measures the size of

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... 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.