We present the results of a microscopic calculation of the longitudinal the
rmal conductivity, kappa(xx), of a d-wave superconductor in the mixed stale
. Our results show an increase in the thermal conductivity with the applied
field at low temperatures and a decrease followed by a nearly held indepen
dent kappa(xx)(H) at higher temperatures, in qualitative cement with the ex
perimental results. We discuss the relationship between the slope of the su
perconducting gap and the plateau in kappa(xx)(H).