  We present sufficient conditions so that a conformal map between planar
domains whose boundary components are Jordan curves or points has a continuous
or homeomorphic extension to the closures of the domains. Our conditions
involve the notions of cofat domains and CNED sets, i.e., countably negligible
for extremal distance, recently introduced by the author. We use this result
towards establishing conformal rigidity of a class of circle domains. A circle
domain is conformally rigid if every conformal map onto another circle domain
is the restriction of a M\"obius transformation. We show that circle domains
whose point boundary components are CNED are conformally rigid. This result is
the strongest among all earlier works and provides substantial evidence towards
the rigidity conjecture of He-Schramm, relating the problems of conformal
rigidity and removability.
