A description is presented of an experimental study of the growth and propagation of a large‐amplitude slow space‐charge wave on a weakly relativistic electron beam. The wave is grown as a result of an unstable interaction between a slow space‐charge wave on the electron beam, and the TM modes of a periodic waveguide. Following wave growth the modulated beam is propagated through a cylindrical waveguide. Experimental results show that we have grown, and can propagate, a coherent wave, having a peak electric field of 60 kV/cm, with a phase velocity of 0.29c. The results presented have application to the collective acceleration of ions in intense relativistic electron beams.