This paper presents the results of an investigation of the quiet Sun's magnetic field based on high-resolution infrared spectropolarimetric observations obtained with the Tenerife Infrared Polarimeter (TIP) at the German VTT of the Observatorio del Teide. We observed two very quiet regions at disc centre. The seeing was exceptionally good during both observing runs, being excellent during one of them. In both cases the network was intentionally avoided to the extent possible, to focus the analysis on the characteristics of the weak polarization signals of the inter-network regions. We find that the Stokes V profile of Fe I 15648 Å line in almost 50% of the pixels and Stokes Q and/or U in 20% of the pixels have a signal above 10-3 (in units of continuum intensity Ic), which is significantly above the noise level of 2-3 x 10-4. This implies that we detect fluxes as low as 2 x 1015 Mx/px. We find evidence that we have detected most of the net flux that is in principle detectable at 1'' resolution with the Zeeman effect. The observed linear polarization resulting from the transverse Zeeman effect indicates that the magnetic fields have a broad range of inclinations, although most of the pixels show polarization signatures which imply an inclination of about 20o. Nearly 30% of the selected V-profiles have irregular shapes with 3 or more lobes, suggesting mixed polarities with different LOS velocity within the resolution element. The profiles are classified using a single value decomposition approach. The spatial distribution of the magnetic signal shows that profiles of different classes (having different velocities, splitting, asymmetries) are clustered together and form patches, close to the spatial resolution in size. Most of the field is found to be located in intergranular lanes. The statistical properties of the mainly inter-network field sampled by these observations are presented, showing that most of the observed fields are weak with relatively few kG features. The field strength distribution peaks at 350 G and has a FWHM of 300 G. Other parameters, such as profile asymmetries, filling factors and line-of-sight velocities are also determined and discussed. Based on observations with the German Vacuum Tower Telescope (VTT) operated by the Kiepenheuer-Institut für Sonnenphysik at the Spanish Observatorio del Teide of the Instituto de Astrofísica de Canarias (IAC).