  Given a compact Riemannian manifold, with positive Yamabe quotient, not
conformally diffeomorphic to the standard sphere, we prove a priori estimates
for solutions to the Yamabe problem. We restrict ourselves to the dimensions
less than or equal to 7, where the Positive Mass Theorem is known to be true.
We also show that, when the dimension is greater than or equal to 6, the Weyl
tensor has to vanish at a point where solutions to the Yamabe equation blow up.
