Context: Observations made with the spectropolarimeter onboard the HINODE satellite have detected abundant horizontal magnetic fields in the internetwork quiet Sun. Aims: We compare the results for the horizontal fields obtained at 630 nm with ground-based observations at 1.56 μm, where the sensitivity to magnetic fields is higher than in the visible. Methods: We obtained 30-s integrated spectropolarimetric data of the quiet Sun on disc centre during a period of extremely stable and good seeing. The data have a rms noise in polarization of around 2 × 10-4 of the continuum intensity. The low noise level allows the spectra to be inverted with the SIR code. We compare the inversion results with proxies to determine the magnetic flux. Results: We confirm the presence of the horizontal fields in the quiet Sun internetwork as reported for the satellite data, including voids without linear polarization signal that extend over an area of a few granules. Voids in the circular polarization signal are only of granular scale. More than 60% of the surface show polarization signals of above four times the rms noise level. We find that the total magnetic flux contained in the more inclined to horizontal fields (γ > 45°) is lower by a factor of around 2 than that of the less inclined fields. The proxies for flux determination are strongly affected by the thermodynamic state of the atmosphere, and hence, seem to be unreliable. Conclusions: During spells of good seeing conditions, adaptive optics can render ground-based slit-spectrograph observations at a 70-cm telescope equivalent to the seeing-free space-based data of half-meter class telescopes. We suggest that the difference in the ratio of horizontal to transversal flux between the ground-based infrared data and the satellite-based visible data is due to the different formation heights of the respective spectral lines. We emphasize that the true amount of magnetic flux cannot be derived directly from the spectra. For purely horizontal flux, one would need its vertical extension that has to be estimated by explicit modeling, using the observed spectra as boundary conditions, or be taken from MHD simulations. Time-series of the evolution of the magnetic flux and chromospheric diagnostics are needed to address its possible contribution to chromospheric heating. Appendices A and B are only available in electronic form at http://www.aanda.org