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A novel laser-plasma-based source of relativistic electrons is described. It involves a combination of orthogonally directed laser beams, which are focused in a plasma. One beam excites a wakefield electron plasma wave. Another locally alters the trajectory of some of the electrons in such a way that they can be accelerated and trapped by the wave. With currently available table-top terawatt lasers, a single ultrashort-duration electron bunch can be accelerated to multi-MeV energies in a fraction of a millimeter, with femtosecond synchronization between the light pulse, the electron bunch, and the plasma wave. Both analytical and numerical-simulation results are presented.