  It is shown that if $f$ and $g$ are any two analytic critical circle mappings
with the same irrational rotation number, then the conjugacy that maps the
critical point of $f$ to that of $g$ has regularity $C^{1+\alpha}$ at the
critical point, with a universal value of $\alpha>0$. As a consequence, a new
proof of the hyperbolicity of the full renormalization horseshoe of critical
circle maps is given.
