  We compute the conservative piece of the gravitational self-force (GSF)
acting on a particle of mass m_1 as it moves along an (unstable) circular
geodesic orbit between the innermost stable circular orbit (ISCO) and the light
ring of a Schwarzschild black hole of mass m_2>> m_1. More precisely, we
construct the function h_{uu}(x) = h_{\mu\nu} u^{\mu} u^{\nu} (related to
Detweiler's gauge-invariant "redshift" variable), where h_{\mu\nu} is the
regularized metric perturbation in the Lorenz gauge, u^{\mu} is the
four-velocity of m_1, and x= [Gc^{-3}(m_1+m_2)\Omega]^{2/3} is an invariant
coordinate constructed from the orbital frequency \Omega. In particular, we
explore the behavior of h_{uu} just outside the "light ring" at x=1/3, where
the circular orbit becomes null. Using the recently discovered link between
h_{uu} and the piece a(u), linear in the symmetric mass ratio \nu, of the main
radial potential A(u,\nu) of the Effective One Body (EOB) formalism, we compute
a(u) over the entire domain 0<u<1/3. We find that a(u) diverges at the
light-ring as ~0.25 (1-3u)^{-1/2}, explain the physical origin of this
divergence, and discuss its consequences for the EOB formalism. We construct
accurate global analytic fits for a(u), valid on the entire domain 0<u<1/3 (and
possibly beyond), and give accurate numerical estimates of the values of a(u)
and its first 3 derivatives at the ISCO, as well as the O(\nu) shift in the
ISCO frequency. In previous work we used GSF data on slightly eccentric orbits
to compute a certain linear combination of a(u) and its first two derivatives,
involving also the O(\nu) piece \bar d(u) of a second EOB radial potential
{\bar D}(u,\nu). Combining these results with our present global analytic
representation of a(u), we numerically compute {\bar d}(u)$ on the interval
0<u\leq 1/6.
