Weakly relativistic electron-acoustic solitons are investigated in a two-electron-component plasma whose cool electrons form a relativistic beam. A general Korteweg-de Vries (KdV) equation is derived, in the small-|ø| domain, for a plasma consisting of an arbitrary number of relativistically streaming fluid components and a hot Boltzmann component. This equation is then applied to the specific case of electron-acoustic waves. In addition, the fully nonlinear system of fluid and Poisson equations is integrated to yield electron-acoustic solitons of arbitrary amplitude. It is shown that relativistic beam effects on electron-acoustic solitons significantly increase the soliton amplitude beyond its non-relativistic value. For intermediate- to large-amplitude solitons, a finite cool-electron temperature is found to destroy the balance between nonlinearity and dispersion, yielding soliton break-up. Also, only rarefactive electronacoustic soliton solutions of our equations are found, even though the relativistic beam provides a positive contribution to the nonlinear coefficient of the KdV equation, describing relativistic, nonlinear electron-acoustic waves.