We present measurements of [Fe/H] for six M supergiant stars and three giant stars within 0.5 pc of the Galactic Center (GC) and one M supergiant star within 30 pc of the GC. The results are based on high-resolution ( lambda / DELTA lambda = 40,000) K-band spectra, taken with CSHELL at the NASA Infrared Telescope Facility. We determine the iron abundance by detailed abundance analysis, performed with the spectral synthesis program MOOG. The mean [Fe/H] of the GC stars is determined to be near solar, [Fe/H] = +0.12 +/- 0.22. Our analysis is a differential analysis, as we have observed and applied the same analysis technique to eleven cool, luminous stars in the solar neighborhood with similar temperatures and luminosities as the GC stars. The mean [Fe/H] of the solar neighborhood comparison stars, [Fe/H] = +0.03 +/- 0.16, is similar to that of the GC stars. The width of the GC [Fe/H] distribution is found to be narrower than the width of the [Fe/H] distribution of Baade's Window in the bulge but consistent with the width of the [Fe/H] distribution of giant and supergiant stars in the solar neighborhood.