The following variational principle is obtained for the entropy $S(E)$ of a system with energy $E:S(E)\ensuremath{\geqq}\ensuremath{-}k \mathrm{ln}(\mathrm{Trace}{U}^{2})$ for all non-negative Hermitian density matrices $U$ with Trace $U=1$, Trace $HU=E$; $H$ is the Hamiltonian and $k$ is Boltzmann's constant. The equality sign is realized with this principle for the density matrix of the microcanonical ensemble, as well as for a wide class of similar ensembles (in the limit of large volume).