  An approximate form for the vacuum averaged stress-energy tensor of a
conformal spin-2 quantum field on a black hole background is employed as a
source term in the semiclassical Einstein equations. Analytic corrections to
the Schwarzschild metric are obtained to first order in $\epsilon =
{\hbar}/M^2$, where $M$ denotes the mass of the black hole. The approximate
tensor possesses the exact trace anomaly and the proper asymptotic behavior at
spatial infinity, is conserved with respect to the background metric and is
uniquely defined up to a free parameter $\hat c_2$, which relates to the
average quantum fluctuation of the field at the horizon. We are able to
determine and calculate an explicit upper bound on $\hat c_2$ by requiring that
the entropy due to the back-reaction be a positive increasing function in $r$.
A lower bound for $\hat c_2$ can be established by requiring that the metric
perturbations be uniformly small throughout the region $2M \leq r < r_o$, where
$r_o$ is the radius of perturbative validity of the modified metric. Additional
insight into the nature of the perturbed spacetime outside the black hole is
provided by studying the effective potential for test particles in the vicinity
of the horizon.
