A series of measurements on the propagation speed of a He pressure front in a cryogenically cooled 75.3 m long, 4.3 cm i.d. stainless steel tube at 1.9 K and 4.25 K and a theoretical model for the phenomenon of limited propagation speed of such a He pressure front are described. The measurements have shown that it took 20 h for a He leak of 5.7 x 10(-5) Torr l/s to be detected at the other end of the 75.3 m tube at 1.9 K and if the leak was increased to 1.4 x 10(-4) Torr l/s, still at 1.9 K, the time for the He to be detected at the distant end was reduced to 8.7 h. With the tube at 4.25 K, it took 5.0 h for a He leak of 7.0 x 10(-5) Torr l/s to be detected at the other end. The He travels through the tube in the form of a pressure front with a steep leading pressure gradient. A model involving the adsorption isotherm for He on stainless steel which is able to predict the time of arrival of the He pressure front at the distant end within 17% is described.