Measurements of the thermal-conductivity coefficient κ are reported for liquid He4 I between 1.77 and 3.95°K, for fluid He3 between 1.5 and 3.95°K, both at pressures up to 34 atm, and for gaseous He3 and He4 between 1.5 and 3.95°K at ∼10 Torr. Special attention is given the liquid-vapor critical region of He3 and the λ-transition line of He4. Corrections for effects of thermal boundary resistance and convection are discussed for the fixed-separation parallel-plate apparatus used for these experiments. Taking into account these corrections, the over-all accuracy of the data is considered to be better than ±3%, though the precision is better than ±1%. Away from the singular regions (∂κ∂T)P is anomalously positive and increases with pressure for both He3 and He4. Isobars of κ for He4 I pass through shallow minima and then rise sharply as the λ line is approached from higher temperature. Isotherms of κ for He3 in the neighborhood of the critical point display distinct cusps. Scaling laws predict that near the λ temperature Tλ the coefficient κ should be proportional to (T−Tλ)−13, and near the critical temperature Tc it should be proportional to |T−Tc|−23; other theories predict κ to be proportional to |T−Tc|−12 near Tc. The experimental data are found to agree qualitatively, but not quantitatively, with these predictions.