  We present sensitive ALMA observations of TWA 3, a nearby, young ($\sim$10
Myr) hierarchical system composed of three pre-main sequence M3--M4.5 stars.
For the first time, we detected ${}^{12}$CO and ${}^{13}$CO $J$=2-1 emission
from the circumbinary protoplanetary disk around TWA 3A. We jointly fit the
protoplanetary disk velocity field, stellar astrometric positions, and stellar
radial velocities to infer the architecture of the system. The Aa and Ab stars
($0.29\pm0.01\,M_\odot$ and $0.24\pm0.01\,M_\odot$, respectively) comprising
the tight ($P=35$ days) eccentric ($e=0.63\pm0.01$) spectroscopic binary are
coplanar with their circumbinary disk (misalignment $< 6^{\circ}$ with 68%
confidence), similar to other short-period binary systems. From models of the
spectral energy distribution, we found the inner radius of the circumbinary
disk ($r_\mathrm{inner} = 0.50 - 0.75$ au) to be consistent with theoretical
predictions of dynamical truncation $r_\mathrm{cav}/a_\mathrm{inner} \approx
3$. The outer orbit of the tertiary star B ($0.40\pm0.28\,M_\odot$, $a\sim65
\pm 18$ au, $e=0.3\pm0.2$) is not as well constrained as the inner orbit,
however, orbits coplanar with the A system are still preferred (misalignment $
< 20^{\circ}$). To better understand the influence of the B orbit on the TWA 3A
circumbinary disk, we performed SPH simulations of the system and found that
the outer edge of the gas disk ($r_\mathrm{outer}=8.5\pm0.2$ au) is most
consistent with truncation from a coplanar, circular or moderately eccentric
orbit, supporting the preference from the joint orbital fit.
