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We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudo-riemannian manifold $(M,g)$. In other words, we establish a canonical isomorphism between the spaces of polynomials on $T^{*}M$ and of differential operators on tensor densities over $M$, both viewed as modules over the Lie algebra $\mathrm{o}(p+1,q+1)$ where $p+q=dim\left(M\right)$. This quantization exists for generic values of the weights of the tensor densities and we compute the critical values of the weights yielding...