The pressure gradient associated with the flow of heat in helium I and II has been measured using capillary tubes of diameter one-half to 1.1 millimeters. The observations in helium I can be explained as consequences of the vaporization of the liquid in the capillary. In helium II, for heat currents below a certain critical value, which depends on the dimensions of the apparatus and the temperature, the observed pressure gradients agree with the predictions of the thermo-hydrodynamic equations for liquid helium. These equations appear to break down If the critical heat current is exceeded. After this current has been exceeded, the behavior of the pressure gradient in helium II strongly resembles the behavior seen in helium I.