  We prove that two $C^r$ critical circle maps with the same rotation number of
bounded type are $C^{1+\alpha}$ conjugate for some $\alpha>0$ provided their
successive renormalizations converge together at an exponential rate in the
$C^0$ sense. The number $\alpha$ depends only on the rate of convergence. We
also give examples of $C^\infty$ critical circle maps with the same rotation
number that are not $C^{1+\beta}$ conjugate for any $\beta>0$.
