  We present a high-resolution ($R\sim35,000$), high signal-to-noise
($S/N=350$) Magellan/MIKE spectrum of the bright extremely metal-poor star
2MASS~J1808$-$5104. We find [Fe/H] = $-$4.01 (spectroscopic LTE stellar
parameters), [Fe/H] = $-$3.8 (photometric stellar parameters), [Fe/H] = $-$3.7
(spectroscopic NLTE stellar parameters). We measured a carbon-to-iron ratio of
$\mbox{[C/Fe]}= 0.38$ from the CH G-band. J1808$-$5104 is thus not
carbon-enhanced, contrary to many other stars with similarly low iron
abundances. We also determine, for the first time, a barium abundance
($\mbox{[Ba/Fe]} =-0.78$), and obtain a significantly reduced upper limit for
the nitrogen abundance ([N/Fe]$ < - 0.2$). J1808$-$5104 has low ratio of
$\mbox{[Sr/Ba]}=-0.17$, which is consistent with that of stars in ultra-faint
dwarf galaxies. We also fit the abundance pattern of J1808$-$5104 with
nucleosynthesis yields from a grid of Population\,III supernova models. There
is a good fit to the abundance pattern which suggests J1808$-$5104 originated
from gas enriched by a single massive supernova with a high explosion energy of
E $=10\times10^{51}$\,erg and a progenitor stellar mass of
M$=29.5$\,M$_{\odot}$. Interestingly, J1808$-$5104 is a member of the Galactic
thin disk, as confirmed by our detailed kinematic analysis and calculated
stellar actions and velocities. Finally, we also established the orbital
history of J1808$-$5104 using our time-dependent Galactic potential the
\texttt{ORIENT}. J1808$-$5104 appears to have a stable quasi-circular orbit and
been largely confined to the thin disk. This unique orbital history, the star's
very old age ($\sim13.5$\,Gyr), and the low [C/Fe] and [Sr/Ba] ratios suggest
that J1808$-$5104 may have formed at the earliest epoch of the hierarchical
assembly of the Milky Way, and it is most likely associated with the primordial
thin disk.
