The relativistic motion of an electron in the electric field of a solitary wave propagating with constant velocity in a long discharge tube is studied. The intensity of the wave is assumed to be sufficiently high to ionize a gas and to produce a well-conducting plasma behind the trailing edge of the wave. The solution describing one-dimensional relativistic motion of an electron in a soliton-like electric field is presented for the waves of negative and positive polarities. For negative polarity waves. Two different kinds of electron motion corresponding to the particle lag or outrunning with respect to the wave are found. An analytic expression for the energy of the accelerated electron is obtained and analyzed, and the conditions for the maximal efficiency of acceleration are discussed.