We present first results from a spectroscopic survey designed to examine the metallicity and kinematics of individual red giant branch stars in the outer halo of the Andromeda spiral galaxy (M31). This study is based on multislit spectroscopy with the Keck II 10 m telescope and Low Resolution Imaging Spectrograph of the Ca II near-infrared triplet in 99 M31 halo candidates in a field at R = 19 kpc on the southeast minor axis with brightnesses from 20 < I < 22. The spectra are used to isolate M31 halo red giants from foreground Milky Way dwarf stars, faint compact background galaxies, and M31 disk giants. The observed distribution of radial velocities is well fitted by an equal mix of foreground Milky Way dwarf stars, drawn from a standard Galactic model and with velocities v 0 km s-1, and M31 halo giants represented by a Gaussian of width σ ~ 150 km s-1 centered on its systemic velocity of v ≈ -300 km s-1. A secure sample of 29 M31 red giant stars is identified on the basis of radial velocity (v < -220 km s-1) and, in the case of four intermediate-velocity stars (-160 < v < -220 km s-1), broadband B-I color. For this sample of objects, there is rough agreement between the metallicities derived in independent ways: two different calibrations of the Ca II absorption-line strength and a photometric estimate based on fitting model stellar isochrones to an object's location in a (B-I, I) color-magnitude diagram. The [Fe/H] distribution of M31 halo giants has an rms spread of at least 0.6 dex and spans the 2 dex range over which the abundance measurement methods are calibrated. The mean/median metallicity of the M31 halo is about [Fe/H] = -1.9 to -1.1 dex (depending on the details of metallicity calibration and sample selection) and possibly higher: the high-metallicity end of the distribution is poorly constrained by our data since the selection function for the secure M31 sample excludes over 80% of the giants in solar/supersolar metallicity range. Possible reasons are explored for the apparent discrepancy between the mean [Fe/H] found in our spectroscopic survey (corrected for metallicity selection bias) and the slightly higher mean values found in earlier photometric studies. Field halo red giants in M31 appear to be somewhat more metal-rich on average than their Milky Way counterparts. The M31 halo [Fe/H] distribution is comparable to that of M31 globular clusters, Galactic globular clusters, and Local Group dwarf satellite galaxies. The data in this 19 kpc outer halo field are broadly consistent with a scenario in which the halo is built from the accretion of small stellar subsystems. There are four stars in the secure M31 sample that have particularly strong Ca II lines, indicating solar metallicity, at a common velocity of ≈-340 km s-1 close to the galaxy's systemic velocity, similar to what might be expected for M31 disk giants on the minor axis. An extrapolation of the inner disk brightness profile, however, falls far short of accounting for these four stars—the disk would instead have to be very large (Rdisk 80 kpc) and/or warped. More likely, these four stars represent a metal-rich debris trail from a past accretion event in the halo.