  We study the Casimir interaction between perfectly conducting sphere and
plate in the classical limit of high temperatures. By taking the small-distance
expansion of the exact scattering formula, we compute the leading correction to
the Casimir energy beyond the commonly employed proximity force approximation.
We find that for a sphere of radius $R$ at distance $d$ from the plate the
correction is of the form $\ln^2 (d/R)$, in agreement with indications from
recent large-scale numerical computations. We develop a fast-converging
numerical scheme for computing the Casimir interaction to high precision, based
on bispherical partial waves, and we verify that the short-distance formula
provides precise values of the Casimir energy also for fairly large distances.
