  We present high-resolution $^{12}$CO and $^{13}$CO 2-1 ALMA observations, as
well as optical and near-infrared spectroscopy, of the highly-inclined
protoplanetary disk around SSTC2D J163131.2-242627. The spectral type we derive
for the source is consistent with a $\rm 1.2 \, M_{\odot}$ star inferred from
the ALMA observations. Despite its massive circumstellar disk, we find little
to no evidence for ongoing accretion on the star. The CO maps reveal a disk
that is unusually compact along the vertical direction, consistent with its
appearance in scattered light images. The gas disk extends about twice as far
away as both the submillimeter continuum and the optical scattered light. CO is
detected from two surface layers separated by a midplane region in which CO
emission is suppressed, as expected from freeze-out in the cold midplane. We
apply a modified version of the Topographically Reconstructed Distribution
method presented by Dutrey et al. 2017 to derive the temperature structure of
the disk. We find a temperature in the CO-emitting layers and the midplane of
$\sim$33 K and $\sim$20 K at $\rm R<200$ au, respectively. Outside of $\rm
R>200$ au, the disk's midplane temperature increases to $\sim$30 K, with a
nearly vertically isothermal profile. The transition in CO temperature
coincides with a dramatic reduction in the sub-micron and sub-millimeter
emission from the disk. We interpret this as interstellar UV radiation
providing an additional source of heating to the outer part of the disk.
