  Let $f$ and $\tilde{f}$ be two circle diffeomorphisms with a break point,
with the same irrational rotation number of bounded type, the same size of the
break $c$ and satisfying a certain Zygmund type smoothness condition depending
on a parameter $\gamma>2.$ We prove that under a certain condition imposed on
the break size $c$, the diffeomorphisms $f$ and $\tilde{f}$ are
$C^{1+\omega_{\gamma}}$-smoothly conjugate to each other, where
$\omega_{\gamma}(\delta)=|\log \delta|^{-(\gamma/2-1)}.$
