  A circle domain $\Omega$ in the Riemann sphere is conformally rigid if every
conformal map from $\Omega$ onto another circle domain is the restriction of a
M\"{o}bius transformation. We show that circle domains satisfying a certain
quasihyperbolic condition, which was considered by Jones and Smirnov, are
conformally rigid. In particular, H\"{o}lder circle domains and John circle
domains are all conformally rigid. This provides new evidence for a conjecture
of He and Schramm relating rigidity and conformal removability.
