This paper investigates the distribution of linear polarization signals in the quiet-Sun internetwork using ultra-deep spectropolarimetric data. We reduce the noise of the observations as much as is feasible by adding single-slit measurements of the Zeeman-sensitive Fe I 630 nm lines taken by the Hinode spectropolarimeter. The integrated Stokes spectra are employed to determine the fraction of the field of view covered by linear polarization signals. We find that up to 69% of the quiet solar surface at disk center shows Stokes Q or U profiles with amplitudes larger than 0.032% (4.5 times the noise level of 7 × 10-5 reached by the longer integrations). The mere presence of linear polarization in most of the quiet Sun implies that the weak internetwork fields must be highly inclined, but we quantify this by inverting those pixels with Stokes Q or U signals well above the noise. This allows for a precise determination of the field inclination, field strength, and field azimuth because the information carried by all four Stokes spectra is used simultaneously. The inversion is performed for 53% of the observed field of view at a noise level of 1.3 × 10-4 I c. The derived magnetic distributions are thus representative of more than half of the quiet-Sun internetwork. Our results confirm the conclusions drawn from previous analyses using mainly Stokes I and V: internetwork fields are very inclined, but except in azimuth they do not seem to be isotropically distributed.