  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 x
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 x 10^{-4}. 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.
