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 (five RGs) or in fiber mode (39 RGs, FLAMES/GIRAFFE-UVES). The sample is more than a factor of 4 larger than any previous spectroscopic investigation of stars in dSphs based on high-resolution (R≥38000) 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. For the 27 stars for which we measured both Fe I and Fe II abundances, we found evidence of NLTE effects between neutral and singly ionized iron abundances. The difference is ˜0.1 dex, on average, but steadily increases when moving from the metal-rich to the metal-poor regime. Moreover, the two metallicity distributions differ at the 97% confidence level. Assuming that the Fe II abundances are minimally affected by NLTE effects, we corrected the Fe I stellar abundances using a linear fit between Fe I and Fe II stellar abundance determinations. We found that the Carina metallicity distribution based on the corrected Fe I abundances (44 RGs) has a weighted mean metallicity of [Fe/H] = -1.80 and a weighted standard deviation of σ = 0.24 dex. The Carina metallicity distribution based on the Fe II abundances (27 RGs) gives similar estimates ([Fe/H] = -1.72, σ = 0.24 dex). The current weighted mean metallicities are slightly more metal-poor when compared with similar estimates available in the literature. Furthermore, when we restricted our analysis to stars with the most accurate iron abundances, ˜20 Fe I and at least three Fe II measurements (15 stars), we found that the range in iron abundances covered by Carina RGs (˜1 dex) agreed quite well with similar estimates based on high-resolution spectra; however, it is a factor of 2-3 smaller than abundance estimates based on the near-infrared calcium triplet. This finding supports previous estimates based on photometric metallicity indicators.