  We compute $O(\alpha'^3)$ corrections to the $AdS_5\times S^5$ black hole
metric. We find that the radius of the $S^5$ depends on the radial $AdS_5$
coordinate. This completes the computation of Gubser, Klebanov and Tseytlin
(hep-th/9805156). The fact that the metric no longer factorizes should modify
the value of the Wilson line at finite temperature and the glueball mass
spectrum.
