The General Theory of Relativity predicts the redshift of spectral lines in the solar photosphere, as a consequence of the gravitational potential of the Sun. This effect can be measured from a solar disk-integrated flux spectrum of the Suns reflected light on solar system bodies. The laser frequency comb (LFC) calibration system attached to the HARPS spectrograph offers the possibility to perform an accurate measurement of the solar gravitational redshift (GRS) by observing the Moon or other solar system bodies. We have analysed the line shift observed in Fe absorption lines from five high-quality HARPS-LFC spectra of the Moon. We select an initial sample of 326 photospheric Fe lines in the spectral range 476-585 nm and measure their line positions and equivalent widths (EWs). Accurate line shifts are derived from the wavelength position of the core of the lines compared with the laboratory wavelengths. We fit the observed spectral Fe lines using CO$^5$BOLD 3D synthetic profiles. Convective motions in the solar photosphere do not affect the line cores of Fe lines stronger than about $sim 150$ mA. In our sample, only 15 FeI lines have EWs in the range $150 <$ EW(mA) $< 550$, providing a measurement of the solar GRS at $639pm14$ ${rm m;s^{-1}}$, consistent with the expected theoretical value on Earth of $sim 633.1$ ${rm m;s^{-1}}$. A final sample of about 97 weak Fe lines with EW $<180$ mA allows us to derive a mean global line shift of $638pm6$ ${rm m;s^{-1}}$ in agreement with the theoretical solar GRS. These are the most accurate measurements of the solar GRS so far. Ultrastable spectrographs calibrated with the LFC over a larger spectral range, such as HARPS or ESPRESSO, together with a further improvement on the laboratory wavelengths, could provide a more robust measurement of the solar GRS and further tests for the 3D hydrodynamical models.
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