Tamassy and Binh introduced weakly symmetric Riemannian manifolds. The properties of weakly symmetric and weakly Ricci symmetric Riemannian manifolds are studied by some authors. In this paper, the conformally flat weakly Ricci symmetric manifolds is considered. In this case the Ricci tensor of Riemannian manifold is a quadratic Killing tensor, and some properties of this manifold are obtained. In conclusion, it is found that the energy momentum tensor of this space in a perfect fluid is a quasi-Einstein tensor and also a Codazzi tensor.