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 ($0.29pm0.01,M_odot$ and $0.24pm0.01,M_odot$, respectively) comprising the tight ($P=35$ days) eccentric ($e=0.63pm0.01$) 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 ($0.40pm0.28,M_odot$, $asim65 pm 18$ au, $e=0.3pm0.2$) 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}=8.5pm0.2$ au) is most consistent with truncation from a coplanar, circular or moderately eccentric orbit, supporting the preference from the joint orbital fit.
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