  Using a recently presented numerical code for calculating the Lorenz-gauge
gravitational self-force (GSF), we compute the $O(m)$ conservative correction
to the precession rate of the small-eccentricity orbits of a particle of mass
$m$ moving around a Schwarzschild black hole of mass ${\mathsf M}\gg m$.
Specifically, we study the gauge-invariant function $\rho(x)$, where $\rho$ is
defined as the $O(m)$ part of the dimensionless ratio
$(\hat\Omega_r/\hat\Omega_{\varphi})^2$ between the squares of the radial and
azimuthal frequencies of the orbit, and where $x=[Gc^{-3}({\mathsf
M}+m)\hat\Omega_{\varphi}]^{2/3}$ is a gauge-invariant measure of the
dimensionless gravitational potential (mass over radius) associated with the
mean circular orbit. Our GSF computation of the function $\rho(x)$ in the
interval $0<x\leq 1/6$ determines, for the first time, the {\em strong-field
behavior} of a combination of two of the basic functions entering the Effective
One Body (EOB) description of the conservative dynamics of binary systems. We
show that our results agree well in the weak-field regime (small $x$) with the
3rd post-Newtonian (PN) expansion of the EOB results, and that this agreement
is improved when taking into account the analytic values of some of the
logarithmic-running terms occurring at higher PN orders. Furthermore, we
demonstrate that GSF data give access to higher-order PN terms of $\rho(x)$ and
can be used to set useful new constraints on the values of yet-undetermined EOB
parameters. Most significantly, we observe that an {\em excellent global
representation} of $\rho(x)$ can be obtained using a simple `two-point'
Pad\'{e} approximant which combines 3PN knowledge at $x=0$ with GSF information
at a single strong-field point (say, $x=1/6$).
