High-power lasers that fit into a university-scale laboratory can now reach focused intensities of more than 10(19) W cm(-2) at high repetition rates. Such lasers are capable of producing beams of energetic electrons, protons and gamma-rays. Relativistic electrons are generated through the breaking of large-amplitude relativistic plasma waves created in the wake of the laser pulse as it propagates through a plasma, or through a direct interaction between the laser field and the electrons in the plasma. However, the electron beams produced from previous laser-plasma experiments have a large energy spread, limiting their use for potential applications. Here we report high-resolution energy measurements of the electron beams produced from intense laser-plasma interactions, showing that--under particular plasma conditions--it is possible to generate beams of relativistic electrons with low divergence and a small energy spread (less than three per cent). The monoenergetic features were observed in the electron energy spectrum for plasma densities just above a threshold required for breaking of the plasma wave. These features were observed consistently in the electron spectrum, although the energy of the beam was observed to vary from shot to shot. If the issue of energy reproducibility can be addressed, it should be possible to generate ultrashort monoenergetic electron bunches of tunable energy, holding great promise for the future development of 'table-top' particle accelerators.