  Carbon $1s$ core-hole excitation of the molecular anion C$_2^-$ has been
experimentally studied at high resolution by employing the photon-ion
merged-beams technique at a synchrotron light source. The experimental cross
section for photo--double-detachment shows a pronounced vibrational structure
associated with $1\sigma_u\to3\sigma_g$ and $1\sigma_g \to 1\pi_u$ core
excitations of the C$_2^-$ ground level and first excited level, respectively.
A detailed Franck-Condon analysis reveals a strong contraction of the C$_2^-$
molecular anion by 0.2~\AA\ upon this core photoexcitation. The associated
change of the molecule's moment of inertia leads to a noticeable rotational
broadening of the observed vibrational spectral features. This broadening is
accounted for in the present analysis which provides the spectroscopic
parameters of the C$_2^-$ $1\sigma_u^{-1}\,3\sigma_g^2\;{^2}\Sigma_u^+$ and
$1\sigma_g^{-1}\,3\sigma_g^2\;{^2}\Sigma_g^+$ core-excited levels.
