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We present the first results from the HST Archival Legacy project "SKYSURF." As described in Windhorst et al. 2022, SKYSURF utilizes the large HST archive to study the diffuse UV, optical, and near-IR backgrounds and foregrounds in detail. Here we utilize SKYSURF's first sky-surface brightness measurements to constrain the level of near-IR diffuse Extragalactic Background Light (EBL). Our sky-surface brightness measurements have been verified to an accuracy of better than 1%, which when combined with systematic errors associated with HST, results in sky brightness uncertainties of $\sim$2-4% $\simeq$ 0.005 MJy/sr in each image. We put limits on the amount of diffuse EBL in three near-IR filters (F125W, F140W, and F160W) by comparing our preliminary sky measurements of $> 30,000$ images to Zodiacal light models, carefully selecting the darkest images to avoid contamination from stray light. In addition, we investigate the impact that instrumental thermal emission has on our measurements, finding that it has a limited impact on F125W and F140W measurements, whereas uncertainties in the exact thermal state of HST results in significant uncertainties in the level of astrophysical diffuse light in F160W images. When compared to the Kelsall et al. (1998) Zodiacal model, an isotropic diffuse background of $30$ nW m$^{-2}$ sr$^{-1}$ remains, whereas using the Wright (1998) Zodiacal model results in no discernible diffuse background. Based primarily on uncertainties in the foreground model subtraction, we present limits on the amount of diffuse EBL of 29 nW m$^{-2}$ sr$^{-1}$, 40 nW m$^{-2}$ sr$^{-1}$, and 29 nW m$^{-2}$ sr$^{-1}$ for F125W, F140W, and F160W respectively. While this light is generally isotropic, our modeling at this point does not distinguish between a cosmological origin or a Solar System origin (such as a dim, diffuse, spherical cloud of cometary dust)., Accepted with Windhorst et al. 2022 to AJ. Main figures are Fig. 10 and 11. Comments welcome!