  We study conservative finite-mass corrections to the motion of a particle in
a bound (eccentric) strong-field orbit around a Schwarzschild black hole. We
assume the particle's mass $\mu$ is much smaller than the black hole mass $M$,
and explore post-geodesic corrections of $O(\mu/M)$. Our analysis uses
numerical data from a recently developed code that outputs the Lorenz-gauge
gravitational self-force (GSF) acting on the particle along the eccentric
geodesic. First, we calculate the $O(\mu/M)$ conservative correction to the
periastron advance of the orbit, as a function of the (gauge-dependent)
semilatus rectum and eccentricity. A gauge-invariant description of the GSF
precession effect is made possible in the circular-orbit limit, where we
express the correction to the periastron advance as a function of the invariant
azimuthal frequency. We compare this relation with results from fully nonlinear
numerical-relativistic simulations. In order to obtain a gauge-invariant
measure of the GSF effect for fully eccentric orbits, we introduce a suitable
generalization of Detweiler's circular-orbit "redshift" invariant. We compute
the $O(\mu/M)$ conservative correction to this invariant, expressed as a
function of the two invariant frequencies that parametrize the orbit. Our
results are in good agreement with results from post-Newtonian calculations in
the weak-field regime, as we shall report elsewhere. The results of our study
can inform the development of analytical models for the dynamics of strongly
gravitating binaries. They also provide an accurate benchmark for future
numerical-relativistic simulations.
