Observations of the quiet Sun observed with the Advanced Stokes Polarimeter are used to explore the distribution of apparent flux density (Bapp) and size scales of internetwork regions. For a typical disk-center quiet-Sun observation with excellent seeing (at a quantifiable angular resolution of 1''), average |Bapp| of 18.6 and 10.7 Mx cm-2 are found for the entire quiet region map and the internetwork region (2 σ noise < |Bapp| < 40 Mx cm-2), respectively. The weak internetwork flux appears to consist of two components: the spatially concentrated "granular" internetwork fields and a more diffuse, weaker component that has a characteristic size scale of a few arcseconds. Most of the internetwork area is occupied by measurable fields: 69% of the area has apparent flux density greater than 4.5 Mx cm-2 (3 σ), or 84% with flux density greater than 3.0 Mx cm-2 (2 σ). If the results of Hanle depolarization measurements are accepted, the internetwork flux detected here must be close (within a factor of 2-3) to being spatially resolved and must be intrinsically weak (10-30 G). Examination of Stokes polarimetry in quiet regions away from disk center indicates that the central cores of network flux, as indicated by high-polarization signals, do not have a high degree of mixed polarity. In contrast, regions immediately surrounding the network elements have stronger linear polarization than would be expected from locally vertical fields, indicating a high degree of mixed polarity. This mixed polarity may be the result of the continual "sweeping" of mixed polarity internetwork flux toward the network boundaries. When a quiet region has a significant imbalance of flux of opposite polarities, the internetwork imbalance is of the same sign and typicallyimg1.gif that of the network. The smaller imbalance in the internetwork suggests that a local dynamo produces most of the internetwork flux.