  We present the design and observations of low resolution JWST/NIRSpec PRISM
spectroscopy from the Ultradeep NIRSpec and NIRCam ObserVations before the
Epoch of Reionization (UNCOVER) Cycle 1 JWST Treasury program. Targets are
selected using JWST/NIRCam photometry from UNCOVER and other programs, and
cover a wide range of categories and redshifts to ensure the legacy value of
the survey. These categories include the first galaxies at $z\gtrsim10$, faint
galaxies during the Epoch of Reionization ($z\gtrsim6-8$), high redshift AGN
($z\gtrsim6$), Population III star candidates, distant quiescent and dusty
galaxies ($1\lesssim z \lesssim 6$), and filler galaxies sampling
redshift--color--magnitude space from $z\sim 0.1-13$. Seven NIRSpec MSA masks
across the extended Abell 2744 cluster were observed, along with NIRCam
parallel imaging in 8 filters (F090W, F115W, F150W, F200W, F277W, F356W, F410M,
F444W, F480M) over a total area of ~26 arcmin$^2$, overlapping existing HST
coverage from programs including the Hubble Frontier Fields and BUFFALO. We
successfully observed 553 objects down to
$m_{\mathrm{F444W}}\sim30\mathrm{AB}$, and by leveraging mask overlaps, we
reach total on-target exposure times ranging from 2.4-16.7h. We demonstrate the
success rate and distribution of confirmed redshifts, and also highlight the
rich information revealed by these ultradeep spectra for a subset of our
targets. An updated lens model of Abell 2744 is also presented, including 14
additional spectroscopic redshifts and finding a total cluster mass of
$M_{\mathrm{SL}}=(2.1\pm0.3)\times10^{15}\,\mathrm{M}_{\odot}$. We publicly
release reduced 1D and 2D spectra for all objects observed in Summer 2023 along
with a spectroscopic redshift catalog and the updated lens model of the cluster
(https://jwst-uncover.github.io/DR4.html).
