  We present Keck-I MOSFIRE spectroscopy in the Y and H bands of GDN-8231, a
massive, compact, star-forming galaxy (SFG) at a redshift $z\sim1.7$. Its
spectrum reveals both H$_{\alpha}$ and [NII] emission lines and strong Balmer
absorption lines. The H$_{\alpha}$ and Spitzer MIPS 24 $\mu$m fluxes are both
weak, thus indicating a low star formation rate of SFR $\lesssim5-10$
M$_{\odot}$ yr$^{-1}$. This, added to a relatively young age of $\sim700$ Myr
measured from the absorption lines, provides the first direct evidence for a
distant galaxy being caught in the act of rapidly shutting down its star
formation. Such quenching allows GDN-8231 to become a compact, quiescent
galaxy, similar to 3 other galaxies in our sample, by $z\sim1.5$. Moreover, the
color profile of GDN-8231 shows a bluer center, consistent with the predictions
of recent simulations for an early phase of inside-out quenching. Its
line-of-sight velocity dispersion for the gas, $\sigma^{\rm{gas}}_{\!_{\rm
LOS}}=127\pm32$ km s$^{-1}$, is nearly 40% smaller than that of its stars,
$\sigma^{\star}_{\!_{\rm LOS}}=215\pm35$ km s$^{-1}$. High-resolution
hydro-simulations of galaxies explain such apparently colder gas kinematics of
up to a factor of $\sim1.5$ with rotating disks being viewed at different
inclinations and/or centrally concentrated star-forming regions. A clear
prediction is that their compact, quiescent descendants preserve some remnant
rotation from their star-forming progenitors.
