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The diffuse, unresolved sky provides most of the photons that the Hubble Space Telescope (HST) receives, yet remains poorly understood. HST Archival Legacy program SKYSURF aims to measure the 0.2-1.6 $\mu$m sky surface brightness (sky-SB) from over 140,000 HST images. We describe a sky-SB measurement algorithm designed for SKYSURF that is able to recover the input sky-SB from simulated images to within 1% uncertainty. We present our sky-SB measurements estimated using this algorithm on the entire SKYSURF database. Comparing our sky-SB spectral energy distribution (SED) to measurements from the literature shows general agreements. Our SKYSURF SED also reveals a possible dependence on Sun angle, indicating either non-isotropic scattering of solar photons off interplanetary dust or an additional component to Zodiacal Light. Finally, we update Diffuse Light limits in the near-IR based on the methods from Carleton et al. (2022), with values of 0.009 MJy sr$^{-1}$ (22 nW m$^{-2}$ sr$^{-1}$) at 1.25 $\mu$m, 0.015 MJy sr$^{-1}$ (32 nW m$^{-2}$ sr$^{-1}$) at 1.4 $\mu$m, and 0.013 MJy sr$^{-1}$ (25 nW m$^{-2}$ sr$^{-1}$) at 1.6 $\mu$m. These estimates provide the most stringent all-sky constraints to date in this wavelength range. SKYSURF sky-SB measurements are made public on the official SKYSURF website and will be used to constrain Diffuse Light in future papers., Comment: Revised based on helpful comments from the reviewer, and accepted to AJ on April 12th, 2023. Main paper: 18 pages, 9 figures, 4 tables. Appendices: 16 pages, 10 figures, 1 table. Main results shown in Figure 7 and Table 4