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Using a recently presented numerical code for calculating the Lorenz-gaugegravitational self-force (GSF), we compute the $O(m)$ conservative correctionto 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 $\rho$ isdefined as the $O(m)$ part of the dimensionless ratio$(\hat\Omega_r/\hat\Omega_{\varphi})^2$ between the squares of the radial andazimuthal frequencies of the orbit, and where $x=[Gc^{-3}({\mathsfM}+m)\hat\Omega_{\varphi}]^{2/3}$ is a gauge-invariant measure of thedimensionless gravitational potential (mass over radius) associated with themean circular orbit. Our GSF computation of the function $\rho(x)$ in theinterval $0<x\leq 1/6$ determines, for the first time, the {\em strong-fieldbehavior} of a combination of two of the basic functions entering the EffectiveOne Body (EOB) description of the conservative dynamics of binary systems. Weshow that our results agree well in the weak-field regime (small $x$) with the3rd post-Newtonian (PN) expansion of the EOB results, and that this agreementis improved when taking into account the analytic values of some of thelogarithmic-running terms occurring at higher PN orders. Furthermore, wedemonstrate that GSF data give access to higher-order PN terms of $\rho(x)$ andcan be used to set useful new constraints on the values of yet-undetermined EOBparameters. Most significantly, we observe that an {\em excellent globalrepresentation} of $\rho(x)$ can be obtained using a simple `two-point'Pad\'{e} approximant which combines 3PN knowledge at $x=0$ with GSF informationat a single strong-field point (say, $x=1/6$).

Precession effect of the gravitational self-force in a Schwarzschild spacetime and the effective one-body formalism