A steady-state method was developed and used to measure the flow rate of saturated liquid-helium films at small pressure heads. In this method a steady heat input into a glass vessel partially immersed in liquid helium generates the film flow inwards. A steady state is established when the rate of film flow is equal to the rate of distillation out of the vessel. The flow rate was derived from the temperature difference and level difference between the liquid in the vessel and the bath. Measurements in the vicinity of 1.2°K indicate a marked pressure-head dependence of the flow rate below a pressure head of 0.01 cm of liquid helium. A maximum flow rate for zero pressure head (0.001 cm of liquid helium) of about 5×10-5 cc/sec cm is observed. This rate is suggested to be the true critical flow rate of the film. © 1962 The American Physical Society.