We present spectroscopic observations of red giant branch ( RGB ) stars over a large expanse in the halo of the Andromeda spiral galaxy ( M31 ) , acquired with the DEIMOS instrument on the Keck II 10-m telescope . Using a combination of five photometric/spectroscopic diagnostics — ( 1 ) radial velocity , ( 2 ) intermediate-width DDO 51 photometry , ( 3 ) Na i equivalent width ( surface gravity sensitive ) , ( 4 ) position in the color-magnitude diagram , and ( 5 ) comparison between photometric and spectroscopic [ Fe/H ] estimates — we isolate over 250 bona fide M31 bulge and halo RGB stars located in twelve fields ranging from R = 12–165 kpc from the center of M31 ( 47 of these stars are halo members with R > 60 kpc ) . We derive the metallicity distribution function of M31 RGB stars in each of these fields by comparing the stellar location in the ( I , V - I ) color-magnitude diagram to a finely spaced grid of theoretical isochrones . The mean of the resulting M31 spheroid ( bulge and halo ) metallicity distribution is found to be systematically more metal-poor with increasing radius , shifting from \langle [ Fe/H ] \rangle = { - } 0.47 \pm 0.03 ( \sigma = 0.39 ) at R < 20 kpc to \langle [ Fe/H ] \rangle = { - } 0.94 \pm 0.06 ( \sigma = 0.60 ) at R \sim 30 kpc to \langle [ Fe/H ] \rangle = { - } 1.26 \pm 0.10 ( \sigma = 0.72 ) at R > 60 kpc , assuming [ \alpha /Fe ] = 0.0 . These results indicate the presence of a metal-poor RGB population at large radial distances out to at least R = 160 kpc , thereby supporting our recent discovery of a stellar halo in M31 : its halo and bulge ( defined as the structural components with R ^ { -2 } power law and de Vaucouleurs R ^ { 1 / 4 } law surface brightness profiles , respectively ) are shown to have distinct metallicity distributions . If we assume an \alpha -enhancement of [ \alpha /Fe ] = +0.3 for M31 ’ s halo , we derive \langle [ Fe/H ] \rangle = { - } 1.5 \pm 0.1 ( \sigma = 0.7 ) . Therefore , the mean metallicity and metallicity spread of this newly found remote M31 RGB population are similar to those of the Milky Way halo .