  We study the regularity of a conjugacy $H$ between a hyperbolic toral
automorphism $A$ and its smooth perturbation $f$ We show that if $H$ is weakly
differentiable then it is $C^{1+H\"older}$ and, if $A$ is also weakly
irreducible, then $H$ is $C^\infty$. As a part of the proof, we establish
results of independent interest on H\"older continuity of a measurable
conjugacy between linear cocycles over a hyperbolic system. As a corollary, we
improve regularity of the conjugacy to $C^\infty$ in prior local rigidity
results.
