  A detailed description of an original method used to measure the luminosity
accumulated by the HERA-B experiment for a data sample taken during the
2002-2003 HERA running period is reported. We show that, with this method, a
total luminosity measurement can be achieved with a typical precision,
including overall systematic uncertainties, at a level of 5% or better. We also
report evidence for the detection of delta-rays generated in the target and
comment on the possible use of such delta rays to measure luminosity.
