  The post-Newtonian gravitoelectric secular rate of the mean anomaly M is
worked out for a two-body system in the framework of the General Theory of
Relativity. The possibility of using such an effect, which is different from
the well known decrease of the orbital period due to gravitational wave
emission, as a further post-Keplerian parameter in binary systems including one
pulsar is examined. The resulting effect is almost three times larger than the
periastron advance \dot\omega. E.g., for the recently discovered double pulsar
system PSR J0737-3039 A+B it would amount to -47.79 deg yr^-1. This implies
that it could be extracted from the linear part of a quadratic fit of the
orbital phase because the uncertainties both in the linear drift due to the
mean motion and in the quadratic shift due to the gravitational wave are
smaller. The availability of such additional post-Keplerian parameter would be
helpful in further constraining the General Theory of Relativity, especially
for such systems in which some of the other post-Keplerian parameters can be
measured with limited accuracy. Moreover, also certain pulsar-white dwarf
binary systems, characterized by circular orbits like PSR B1855+09 and a
limited number of measured post-Keplerian parameters, could be used for
constraining competing theories of gravity.
