We have performed accurate iron abundance measurements for 44 red giants (RGs) in the Carina dwarf spheroidal (dSph) galaxy. We used archival, high-resolution spectra (R~38,000) collected with UVES at ESO/VLT either in slit mode (5) or in fiber mode (39, FLAMES/GIRAFFE-UVES). The sample is more than a factor of four larger than any previous spectroscopic investigation of stars in dSphs based on high-resolution (R>38,000) spectra. We did not impose the ionization equilibrium between neutral and singly-ionized iron lines. The effective temperatures and the surface gravities were estimated by fitting stellar isochrones in the V, B-V color-magnitude diagram. To measure the iron abundance of individual lines we applied the LTE spectrum synthesis fitting method using MARCS model atmospheres of appropriate metallicity. We found evidence of NLTE effects between neutral and singly-ionized iron abundances. Assuming that the FeII abundances are minimally affected by NLTE effects, we corrected the FeI stellar abundances using a linear fit between FeI and FeII stellar abundance determinations. We found that the Carina metallicity distribution based on the corrected FeI abundances (44 RGs) has a weighted mean metallicity of [Fe/H]=-1.80 and a weighted standard deviation of sigma=0.24 dex. The Carina metallicity distribution based on the FeII abundances (27 RGs) gives similar estimates ([Fe/H]=-1.72, sigma=0.24 dex). The current weighted mean metallicities are slightly more metal poor when compared with similar estimates available in the literature. Furthermore, if we restrict our analysis to stars with the most accurate iron abundances, ~20 FeI and at least three FeII measurements (15 stars), we found that the range in iron abundances covered by Carina RGs (~1 dex) agrees quite well with similar estimates based on high-resolution spectra. ; Comment: Accepted for publication in PASP, 16 pages, 7 figures, 3 tables, 1 MR table Note: the electronic version of Table1 is included, but commented, in the tex file