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We have obtained deep images in $UBRI$, using the KPNO 4-m telescope and a $2048\times2048$ CCD, of a $15'\times15'$ field in the outskirts of the Andromeda galaxy (M31). A new technique is used to isolate faint stars from distant field galaxies on the basis of broadband colors and object morphology (i.e.\ whether or not an object is broader than the PSF)---this is much more efficient than using morphology alone (Gould, Guhathakurta, Richstone \& Flynn 1992, ApJ, 388, 345). The technique has been used to identify a sample of individual red giant branch stars at the distance of M31, located about 20$\,$kpc from its center (in projection). A color-magnitude diagram is presented for these stars which allows one to study the shape of the red giant branch in the outer spheroid of M31. The M31 giant branch is compared to the giant branches of a sample of Galactic globular clusters, spanning a wide range of known metallicities, for which data in the same filter combination have been obtained ($UBRI$). The metallicity of the M31 outer spheroid is derived by interpolation between the globular cluster giant branches. In conjunction with the metallicity measurement at 8.6$\,$kpc in M31's spheroid (Pritchet \& van den Bergh, 1988, ApJ, 331, 135), our new measurement permits determination of the metallicity gradient of M31. This quantity is important for understanding galactic evolution. In particular, it can distinguish between the dissipational collapse model (Larson, 1974, MNRAS, 166, 585), which predicts a strong metallicity gradient because of chemical enrichment during collapse, and the accretion model (Searle \& Zinn, 1978, ApJ, 225, 357), which predicts no strong gradient because star formation largely precedes assembly of the galactic spheroid.
