  JWST is revealing a new population of dust-reddened broad-line active
galactic nuclei (AGN) at redshifts $z\gtrsim5$. Here we present deep
NIRSpec/Prism spectroscopy from the Cycle 1 Treasury program UNCOVER of 15 AGN
candidates selected to be compact, with red continua in the rest-frame optical
but with blue slopes in the UV. From NIRCam photometry alone, they could have
been dominated by dusty star formation or AGN. Here we show that the majority
of the compact red sources in UNCOVER are dust-reddened AGN: $60\%$ show
definitive evidence for broad-line H$\alpha$ with FWHM$\, >2000$ km/s, for
$20\%$ current data are inconclusive, and $20\%$ are brown dwarf stars. We
propose an updated photometric criterion to select red $z>5$ AGN that excludes
brown dwarfs and is expected to yield $>80\%$ AGN. Remarkably, among all
$z_{\rm phot}>5$ galaxies with F277W$-$F444W$>1$ in UNCOVER at least $33\%$ are
AGN regardless of compactness, climbing to at least $80\%$ AGN for sources with
F277W$-$F444W$>1.6$. The confirmed AGN have black hole masses of $10^7-10^9$
M$_{\odot}$. While their UV-luminosities ($-16>M_{\rm UV}>-20$ AB mag) are low
compared to UV-selected AGN at these epochs, consistent with percent-level
scattered AGN light or low levels of unobscured star formation, the inferred
bolometric luminosities are typical of $10^7-10^9$ M$_{\odot}$ black holes
radiating at $\sim 10-40\%$ of Eddington. The number densities are surprisingly
high at $\sim10^{-5}$ Mpc$^{-3}$ mag$^{-1}$, 100 times more common than the
faintest UV-selected quasars, while accounting for $\sim1\%$ of the UV-selected
galaxies. While their UV-faintness suggest they may not contribute strongly to
reionization, their ubiquity poses challenges to models of black hole growth.
