We demonstrated experimentally the formation of monoenergetic beams of accelerated electrons by focusing femtosecond laser radiation with an intensity of onto the edge of an aluminum foil. The electrons had energy distributions peaking in the 0.2–0.8 MeV range with energy spread less than 20%. The acceleration mechanism related to the generation of a plasma wave as a result of self-modulation instability of a laser pulse in a dense plasma formed by a prepulse (arriving 12 ns before the main pulse) is considered. One-dimensional and two-dimensional Particle in Cell (PIC) simulations of the laser–plasma interaction showed that effective excitation of a plasma wave as well as trapping and acceleration of an electron beam with an energy on the order of 1 MeV may occur in the presence of sharp gradients in plasma density and in the temporal shape of the pulse.