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We present the discovery of Aquarius III, an ultra-faint Milky Way satellite galaxy identified in the second data release of the DECam Local Volume Exploration (DELVE) survey. Based on deeper follow-up imaging with DECam, we find that Aquarius III is a low-luminosity ($M_V = -2.5^{+0.3}_{-0.5}$; $L_V = 850^{+380}_{-260} \ L_{\odot}$), extended ($r_{1/2} = 41^{+9}_{-8}$ pc) stellar system located in the outer halo ($D_{\odot} = 85 \pm 4$ kpc). From medium-resolution Keck/DEIMOS spectroscopy, we identify 11 member stars and measure a mean heliocentric radial velocity of $v_{\rm sys} = -13.1^{+1.0}_{-0.9} \ \rm km \ s^{-1}$ for the system and place an upper limit of $\sigma_v < 3.5 \rm \ km \ s^{-1}$ ($\sigma_v < 1.6 \rm \ km \ s^{-1}$) on its velocity dispersion at the 95% (68%) credible level. Based on Calcium-Triplet-based metallicities of the six brightest red giant members, we find that Aquarius III is very metal-poor ([Fe/H]$ = -2.61 \pm 0.21$) with a statistically-significant metallicity spread ($\sigma_{\rm [Fe/H]} = 0.46^{+0.26}_{-0.14}$ dex). We interpret this metallicity spread as strong evidence that the system is a dwarf galaxy as opposed to a star cluster. Combining our velocity measurement with $Gaia$ proper motions, we find that Aquarius III is currently situated near its orbital pericenter in the outer halo ($r_{\rm peri} = 78 \pm 7$ kpc) and that it is plausibly on first infall onto the Milky Way. This orbital history likely precludes significant tidal disruption from the Galactic disk, notably unlike other satellites with comparably low velocity dispersion limits in the literature. Thus, if further velocity measurements confirm that its velocity dispersion is truly below $\sigma_v \lesssim 2 \rm \ km \ s^{-1}$, Aquarius III may serve as a useful laboratory for probing galaxy formation physics in low-mass halos.