A line-heat source apparatus was assembled for the purpose of measuring thermal conductivities of particulate samples under low pressures of a carbon dioxide atmosphere. The primary result of this project is the compilation of the first comprehensive suite of measurements of the dependence of thermal conductivity on particle size. The thermal conductivity increases with increasing particle size and atmospheric pressure. In particular, over the range of Martian atmospheric pressures, from 1 to 7 torr, the thermal conductivity was found to be empirically related to approximately the square root of the particle diameter and the square of the cubed root of the atmospheric pressure. At the average pressure of the Martian surface (6 torr) the thermal conductivity varies from 0.011 W/m K, for particles less than 11 μm in diameter, to 0.11 W/m K, for particles 900 μm in diameter. These results differ significantly from the particle size dependence estimated for Mars from previous measurements, except for 200-μm particles, whose thermal conductivity is 0.053 W/m K. The thermal conductivities of larger particles are lower than the previous estimate, by 40% at 900 μm, and the thermal conductivities of smaller particles are higher than the previous estimate, by 60% at 11 μm. These newer estimates agree with other lines of evidence from Martian atmospheric and surficial processes and lead to improved particle size estimates for most of the planet's surface.