  We prove that the operad of framed little 2-discs is formal. Tamarkin and
Kontsevich each proved that the unframed 2-discs operad is formal. The unframed
2-discs is an operad in the category of S^1-spaces, and the framed 2-discs
operad can be constructed from the unframed 2-discs by forming the operadic
semidirect product with the circle group. The idea of our proof is to show that
Kontsevich's chain of quasi-isomorphisms is compatible with the circle actions
and so one can essentially take the operadic semidirect product with the
homology of S^1 everywhere to obtain a chain of quasi-isomorphisms between the
homology and the chains of the framed 2-discs.
