  We report the discovery of Pegasus IV, an ultra-faint dwarf galaxy found in
archival data from the Dark Energy Camera processed by the DECam Local Volume
Exploration Survey. Pegasus IV is a compact, ultra-faint stellar system
($r_{1/2} = 41^{+8}_{-6}$ pc; $M_V = -4.25 \pm 0.2$ mag) located at a
heliocentric distance of $90^{+4}_{-6}$ kpc. Based on spectra of seven
non-variable member stars observed with Magellan/IMACS, we confidently resolve
Pegasus IV's velocity dispersion, measuring $\sigma_{v} = 3.3^{+1.7}_{-1.1}
\text{ km s}^{-1}$ (after excluding three velocity outliers); this implies a
mass-to-light ratio of $M_{1/2}/L_{V,1/2} = 167^{+224}_{-99}
M_{\odot}/L_{\odot}$ for the system. From the five stars with the highest
signal-to-noise spectra, we also measure a systemic metallicity of $\rm [Fe/H]
= -2.67^{+0.25}_{-0.29}$ dex, making Pegasus IV one of the most metal-poor
ultra-faint dwarfs. We tentatively resolve a non-zero metallicity dispersion
for the system. These measurements provide strong evidence that Pegasus IV is a
dark-matter-dominated dwarf galaxy, rather than a star cluster. We measure
Pegasus IV's proper motion using data from Gaia Early Data Release 3, finding
($\mu_{\alpha*}, \mu_{\delta}) = (0.33\pm 0.07, -0.21 \pm 0.08) \text{ mas
yr}^{-1}$. When combined with our measured systemic velocity, this proper
motion suggests that Pegasus IV is on an elliptical, retrograde orbit, and is
currently near its orbital apocenter. Lastly, we identify three potential RR
Lyrae variable stars within Pegasus IV, including one candidate member located
more than ten half-light radii away from the system's centroid. The discovery
of yet another ultra-faint dwarf galaxy strongly suggests that the census of
Milky Way satellites is still incomplete, even within 100 kpc.
