  The nova rate in the Milky Way remains largely uncertain, despite its vital
importance in constraining models of Galactic chemical evolution as well as
understanding progenitor channels for Type Ia supernovae. The rate has been
previously estimated in the range of $\approx10-300$ yr$^{-1}$, either based on
extrapolations from a handful of very bright optical novae or the nova rates in
nearby galaxies; both methods are subject to debatable assumptions. The total
discovery rate of optical novae remains much smaller ($\approx5-10$ yr$^{-1}$)
than these estimates, even with the advent of all-sky optical time domain
surveys. Here, we present a systematic sample of 12 spectroscopically confirmed
Galactic novae detected in the first 17 months of Palomar Gattini-IR (PGIR), a
wide-field near-infrared time domain survey. Operating in $J$-band
($\approx1.2$ $\mu$m) that is relatively immune to dust extinction, the
extinction distribution of the PGIR sample is highly skewed to large extinction
values ($> 50$% of events obscured by $A_V\gtrsim5$ mag). Using recent
estimates for the distribution of mass and dust in the Galaxy, we show that the
observed extinction distribution of the PGIR sample is commensurate with that
expected from dust models. The PGIR extinction distribution is inconsistent
with that reported in previous optical searches (null hypothesis probability $<
0.01$%), suggesting that a large population of highly obscured novae have been
systematically missed in previous optical searches. We perform the first
quantitative simulation of a $3\pi$ time domain survey to estimate the Galactic
nova rate using PGIR, and derive a rate of $\approx 46.0^{+12.5}_{-12.4}$
yr$^{-1}$. Our results suggest that all-sky near-infrared time-domain surveys
are well poised to uncover the Galactic nova population.
