  We prove the validity over $\mathbb{R}$ of a commutative differential graded
algebra model of configuration spaces for simply connected closed smooth
manifolds, answering a conjecture of Lambrechts--Stanley. We get as a result
that the real homotopy type of such configuration spaces only depends on the
real homotopy type of the manifold. We moreover prove, if the dimension of the
manifold is at least $4$, that our model is compatible with the action of the
Fulton--MacPherson operad (weakly equivalent to the little disks operad) when
the manifold is framed. We use this more precise result to get a complex
computing factorization homology of framed manifolds. Our proofs use the same
ideas as Kontsevich's proof of the formality of the little disks operads.
