It is currently accepted that the temperature profile in the fluid outer core of the Earth is adiabatic. As the core is metallic, its thermal conductivity is high and the heat flux conducted down the adiabat can be higher than estimates of the flux leaving the core. The possibility of conductive heat transfer in a thick zone between the convective deep core and the core mantle boundary is envisaged. The conductive regime starts when the heat flux down the adiabat becomes equal to the flux conducted through the thermal boundary layer at the base of the mantle, whose thickness increases with time. Using reasonable values of the physical parameters, it is found that the conductive zone would be about 600 km thick. The possibility of compositional convection causing convection currents in the conductive zone is not excluded.