I recall reading somewhere that if a conformal class contains an Einstein metric then that metric is the unique metric with constant scalar curvature in its conformal class, with the exception of the case of the round sphere. Does this sound right? If it is true: where can I find the proof of this result?

This is not the original reference, but the most general (i.e., applicable to the non-compact and pseudo-Riemannian cases) result I know is:

(The article is publicly available for free.)

Theorem 1* in there gives uniqueness in any case other than simply-connected, constant curvature spaces, and certain warped products with Ricci-flat spaces.

Check out the references in that paper, too.

The first proof of the statement "Einstein metrics are the unique metrics with constant scalar curvature in their conformal class, except for round spheres" is due to Obata in 1971, see MR0303464 M. Obata, The conjectures on conformal transformations of Riemannian manifolds. J. Differential Geometry 6 (1971/72), 247–258. In the beginning of the paper he lists many previous works with partial results in this direction as well.