  We compute the supersymmetric R\'enyi entropies across a spherical
entanglement surface in N=4 SU(N) SYM theory using localization on the
four-dimensional ellipsoid. We extract the leading result at large N and
\lambda, and match its universal part to a gravity calculation involving a
hyperbolically sliced supersymmetric black hole solution of N=4+ SU(2) X U(1)
gauged supergravity in five dimensions. We repeat the analysis in the presence
of a Wilson loop insertion and find again a perfect match with the dual string
theory. Understanding the Wilson loop operator requires knowledge of the full
ten-dimensional IIB supergravity solution which we elaborate upon.
