  We report the results of a pilot program to use the Magellan/M2FS
spectrograph to survey the galactic populations and internal kinematics of
galaxy clusters. For this initial study, we present spectroscopic measurements
for $223$ quiescent galaxies observed along the line of sight to the galaxy
cluster Abell 267 ($z\sim0.23$). We develop a Bayesian method for modeling the
integrated light from each galaxy as a simple stellar population, with free
parameters that specify redshift ($v_\mathrm{los}/c$) and characteristic age,
metallicity ($\mathrm{[Fe/H]}$), alpha-abundance ($[\alpha/\mathrm{Fe}]$), and
internal velocity dispersion ($\sigma_\mathrm{int}$) for individual galaxies.
Parameter estimates derived from our 1.5-hour observation of A267 have median
random errors of $\sigma_{v_\mathrm{los}}=20\ \mathrm{km\ s^{-1}}$,
$\sigma_{\mathrm{Age}}=1.2\ \mathrm{Gyr}$, $\sigma_{\mathrm{[Fe/H]}}=0.11\
\mathrm{dex}$, $\sigma_{[\alpha/\mathrm{Fe}]}=0.07\ \mathrm{dex}$, and
$\sigma_{\sigma_\mathrm{int}}=20\ \mathrm{km\ s^{-1}}$. In a companion paper,
we use these results to model the structure and internal kinematics of A267.
