  A precise comparison of the cross sections for the elastic scattering of
electrons and positrons on protons was performed recently at the VEPP-3 storage
ring in Novosibirsk, Russia. To provide the proposed accuracy, the energies of
the electron and positron beams used during the experiment needed to be
carefully maintained. This paper describes a special beam energy measurement
system used in this experiment. This system is based on measuring the maximum
energy of laser photons backscattered from electron and positron beams. It
provides continuous beam energy measurements with a total relative uncertainty
of less than $4.4 \cdot 10^{-5}$.
