Abstract We describe measurements of the thermal conductivity of a vertical tube partially filled with superfluid liquid helium at saturated vapor pressure. The tube was heated from the top while the bottom was maintained at 0.3 K. Thermal conduction through superfluid film creep and vapor reflux has been modeled in the literature and previously measured at T  ⩾ 0.9 K. From our measurements we assess the validity of the model at the lower temperatures, where significant temperature gradients develop, and consider the effect of ancillary thermal conductances. The work is motivated by the cryogenic requirements for a proposed cryogenic measurement of the electric dipole moment of the neutron.