We present extremely deep Hubble Space Telescope (HST) Wide Field Camera 3 (WFC3) observations of the MUSE Ultra Deep Field. This unique region of the sky contains two quasars at z~3.22 that are separated by only ~500kpc, providing a stereoscopic view of gas and galaxies in emission and absorption across ~10 billion years of cosmic time. We have obtained 90 orbits of HST WFC3 G141 near-infrared grism spectroscopy of this field in a single pointing, as well as 142hr of optical spectroscopy with the Very Large Telescope Multi Unit Spectroscopic Explorer (MUSE). The WFC3 (F140W, F125W, and F336W) and archival WFPC2 (F702W and F450W) imaging provides five-filter photometry that we use to detect 3375 sources between z~0-6, including 1536 objects in a deep central pointing with both spectroscopic and photometric coverage. The F140W and F336W mosaics reach exceptional depths of m_AB_~28 and 29, respectively, providing near-infrared and rest-frame ultraviolet information for 1580 sources, and we reach 5{sigma} continuum detections for objects as faint as m_AB_~27 in the grism spectra. The extensive wavelength coverage of MUSE and WFC3 allows us to measure spectroscopic redshifts for 419 sources, down to galaxy stellar masses of log(M/M_{sun}_)~7 at z~1-2. In this publication, we provide the calibrated HST data and source catalogs as High Level Science Products for use by the community, which includes photometry, morphology, and redshift measurements that enable a variety of studies aimed at advancing our models of galaxy formation and evolution in different environments.