  Improved wavelength calibrators for high-resolution astrophysical
spectrographs will be essential for precision radial velocity (RV) detection of
Earth-like exoplanets and direct observation of cosmological deceleration. The
astro-comb is a combination of an octave-spanning femtosecond laser frequency
comb and a Fabry-P\'erot cavity used to achieve calibrator line spacings that
can be resolved by an astrophysical spectrograph. Systematic spectral shifts
associated with the cavity can be 0.1-1 MHz, corresponding to RV errors of
10-100 cm/s, due to the dispersive properties of the cavity mirrors over broad
spectral widths. Although these systematic shifts are very stable, their
correction is crucial to high accuracy astrophysical spectroscopy. Here, we
demonstrate an \emph{in-situ} technique to determine the systematic shifts of
astro-comb lines due to finite Fabry-P\'erot cavity dispersion. The technique
is practical for implementation at a telescope-based spectrograph to enable
wavelength calibration accuracy better than 10 cm/s.
