The solar photospheric Fe abundance has been determined using realistic ab initio 3D, time-dependent, hydrodynamical model atmospheres. The study is based on the excellent agreement between the predicted and observed line profiles directly rather than equivalent width, since the intrinsic Doppler broadening from the convective motions and oscillations provide the necessary non-thermal broadening. Thus, three of the four hotly debated parameters (equivalent widths, microturbulence and damping enhancement factors) in the center of the recent solar Fe abundance dispute regarding Fe I lines no longer enter the analysis, leaving the transition probabilities as the main uncertainty. Both Fe I (using the samples of lines of both the Oxford and Kiel studies) and Fe II lines have been investigated, which give consistent results: log ǫFeI = 7.44±0.05 and log ǫFeII = 7.45±0.10. Also the wings of strong Fe I lines return consistent abundances, log ǫFeII = 7.42± 0.03, but due to the uncertainties inherent in analyses of strong lines we give this determination lower weight than the results from weak and intermediate strong lines. In view of the recent slight downward revision of the meteoritic Fe abundance log ǫFe = 7.46± 0.01, the agreement between the meteoritic and photospheric values is very good, thus appearingly settling the debate over the photospheric Fe abundance from Fe I lines.