  We present the transmission spectrum of the original transiting hot Jupiter
HD\,209458b from 2.3 -- 5.1 $\mu$m as observed with the NIRCam instrument on
the James Webb Space Telescope (JWST). Previous studies of HD 209458b's
atmosphere have given conflicting results on the abundance of H$_2$O and the
presence of carbon- and nitrogen-bearing species, which have significant
ramifications on the inferences of the planet's metallicity (M/H) and
carbon-to-oxygen (C/O) ratio. We detect strong features of H$_2$O and CO$_2$ in
the JWST transmission spectrum, which when interpreted using a retrieval that
assumes thermochemical equilibrium and fractional grey cloud opacity yields
$3^{+4}_{-1}$ $\times$ solar metallicity and C/O = $0.11^{+0.12}_{-0.06}$. The
derived metallicity is consistent with the atmospheric metallicity-planet mass
trend observed in solar gas giants. The low C/O ratio suggests that this planet
has undergone significant contamination by evaporating planetesimals while
migrating inward. We are also able to place upper limits on the abundances of
CH$_4$, C$_2$H$_2$ and HCN of log($\chi_{\mathrm{CH}_4}$) = -5.6,
log($\chi_{\mathrm{C}_2\mathrm{H}_2}$) = -5.7, and log($\chi_{\mathrm{HCN}}$) =
-5.1, which are in tension with the recent claim of a detection of these
species using ground-based cross-correlation spectroscopy. We find that
HD\,209458b has a weaker CO$_2$ feature size than WASP-39b when comparing their
scale-height-normalized transmission spectra. On the other hand, the size of HD
209458b's H$_2$O feature is stronger, thus reinforcing the low C/O inference.
