  We present Atacama Large Millimeter Array (ALMA) observations of
$^{12}$CO2$-$1 emission from circumstellar material around the massive star
$\eta$~Carinae. These observations reveal new structural details about the cool
equatorial torus located $\sim$4000 au from the star. The CO torus is not a
complete azimuthal loop, but rather, is missing its near side, which appears to
have been cleared away. The missing material matches the direction of apastron
in the eccentric binary system, making it likely that $\eta$~Car's companion
played an important role in disrupting portions of the torus soon after
ejection. Molecular gas seen in ALMA data aligns well with the cool dust around
$\eta$~Car previously observed in mid-infrared (IR) maps, whereas hot dust
resides at the inner surface of the molecular torus. The CO also coincides with
the spatial and velocity structure of near-IR H$_2$ emission. Together, these
suggest that the CO torus seen by ALMA is actually the pinched waist of the
Homunculus polar lobes, which glows brightly because it is close to the star
and warmer than the poles. The near side of the torus appears to be a blowout,
associated with fragmented equatorial ejecta. We discuss implications for the
origin of various features northwest of the star. CO emission from the main
torus implies a total gas mass in the range of 0.2-1 $M_{\odot}$ (possibly up
to 5 $M_{\odot}$ or more, although with questionable assumptions). Deeper
observations are needed to constrain CO emission from the cool polar lobes.
