By means of numerical simulation, we study the scattering of p-polarized light from, and its transmission through, a thin, free-standing, metal film. The illuminated face of the film is a one-dimensional, randomly rough surface, whose generators are perpendicular to the plane of incidence; the back surface is planar. The random roughness of the illuminated surface is characterized by a West-O'Donnell power spectrum that is nonzero in only a narrow range of wave numbers kmin less than k less than kmax that includes the wave numbers q1((omega) ) and q2((omega) ) of the surface plasmon polaritons supported by the film at the frequency (omega) of the incident light. The existence of two surface electromagnetic waves leads to the appearance of two satellite peaks in the angular dependence of the intensity of the incoherent component of the light scattered from the film at scattering angles (theta) s given by sin (theta) s equals - sin (theta) i plus or minus (c/(omega) )[q1((omega) ) - q2((Omega) )], where (theta) i is the angle of incidence of the light, in addition to the enhanced backscattering peak in the retroreflection direction (theta) s equals -(theta) i. At the same time satellite peaks occur in the angular dependence of the intensity of the light transmitted incoherently through the film at angles of transmission (theta) t given by sin(theta) t equals - sin (theta) i plus or minus (c/(omega) )[q1((omega) ) - q2((omega) )], in addition to the enhanced transmission peak in the antispecular direction (theta) t equals -(theta) i. These results are compared with those for a metal film whose rough surface is characterized by a Gaussian power spectrum yielding the same rms height and rms slope as the West-O'Donnell power spectrum.