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>~10, faint galaxies during the Epoch of Reionization (z~6-8), high-redshift active galactic nuclei (z>=6), Population III star candidates, distant quiescent and dusty galaxies (1<~z<~6), and filler galaxies sampling redshift-color-magnitude space from z~0.1-13. Seven NIRSpec microshutter array masks across the extended A2744 cluster were observed, along with NIRCam parallel imaging in nine filters (F090W, F115W, F150W, F200W, F277W, F356W, F410M, F444W, and F480M) over a total area of ~26arcmin2, overlapping existing Hubble Space Telescope coverage from programs including the Hubble Frontier Fields and BUFFALO. We successfully observed 553 objects down to m_F444W_~30 AB, and by leveraging mask overlaps, we reach total on-target exposure times ranging from 2.4 to 16.7hr. We demonstrate the success rate and distribution of the confirmed redshifts, and also highlight the rich information revealed by these ultradeep spectra for a subset of our targets. An updated lens model of A2744 is also presented, including 14 additional spectroscopic redshifts and finding a total cluster mass of M_SL_=(2.1+/-0.3)x10^15^M_{sun}_.