  The effective-one-body theory (EOB) describes the conservative dynamics of
compact binary systems in terms of an effective Hamiltonian approach. The
Hamiltonian for moderately eccentric motion of two non-spinning compact objects
in the extreme mass-ratio limit is given in terms of three potentials: $a(v),
\bar{d}(v), q(v)$. By generalizing the first law of mechanics for
(non-spinning) black hole binaries to eccentric orbits, [\prd{\bf92}, 084021
(2015)] recently obtained new expressions for $\bar{d}(v)$ and $q(v)$ in terms
of quantities that can be readily computed using the gravitational self-force
approach. Using these expressions we present a new computation of the EOB
potential $q(v)$ by combining results from two independent numerical self-force
codes. We determine $q(v)$ for inverse binary separations in the range $1/1200
\le v \lesssim 1/6$. Our computation thus provides the first-ever strong-field
results for $q(v)$. We also obtain $\bar{d}(v)$ in our entire domain to a
fractional accuracy of $\gtrsim 10^{-8}$. We find to our results are compatible
with the known post-Newtonian expansions for $\bar{d}(v)$ and $q(v)$ in the
weak field, and agree with previous (less accurate) numerical results for
$\bar{d}(v)$ in the strong field.
