We demonstrated experimentally the formation of monoenergetic beams of accelerated electrons by focusing femtosecond laser radiation with an intensity of 2 × 10¹⁷ W=cm² 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.