  Symmetric varieties are normal equivarient open embeddings of symmetric
homogeneous spaces, and they are interesting examples of spherical varieties.
We prove that all smooth Fano symmetric varieties with Picard number one admit
K\"{a}hler-Einstein metrics by using a combinatorial criterion for K-stability
of Fano spherical varieties obtained by Delcroix. For this purpose, we present
their algebraic moment polytopes and compute the barycenter of each moment
polytope with respect to the Duistermaat-Heckman measure.
