Advanced Stokes Polarimeter observations are used to study the weakest polarization signals observed in the quiet photosphere with flux densities in the range of 1.5-50 Mx cm-2, which are found in internetwork regions. Our analysis allows us to reach an unprecedented spectropolarimetric sensitivity at the cost of sacrificing spatial resolution. We find evidence for intrinsically different fields in granules and lanes and characterize the average properties of the weakest observable flux concentrations. The magnetic signals observed suggest a strong coupling between magnetic fields and convective flows. Upflows bring up weak fields (equipartition or weaker) to the surface, with stronger upflows carrying larger amounts of flux. The circular polarization profiles observed in the granular regions display a very strongly asymmetric shape, which contrasts with the less asymmetric profiles observed in the downflowing regions. At downflowing locations with speeds of 0.5 km s-1, both weak and strong fields can be found. However, when the downflow speed increases (up to about 1 km s-1) both the mean flux and the intrinsic field strength show a tendency to increase. The asymmetry of the circular polarization profiles also shows a clear trend as a function of magnetic flux density. Low-flux regions display the negative area asymmetry one naturally expects for field strengths decreasing with height embedded in a downflowing environment. As we move to stronger flux density locations, the well-known positive area asymmetry develops and reaches even higher values than those typically found in network regions. These results may have important implications for our understanding of the coupling between magnetic fields and convective processes that pervade the solar photosphere.