  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.
