ABSTRACT We present Northern Extended Millimeter Array (NOEMA) CO(2–1) maps of the z = 0.4418 cluster-central quasi-stellar object (QSO) IRAS 09104+4109, which trace ∼4.5 × 1010 M⊙ of molecular gas in and around the galaxy. As in many low-redshift cool-core clusters, the molecular gas is located in a series of clumps extending along the old radio jets and lobes. It has a relatively low velocity dispersion [336$^{+39}_{-35}$ km s−1 full width at half-maximum (FWHM)] and shows no velocity gradients indicative of outflow or infall. Roughly half the gas is located in a central clump on the north-east side of the galaxy, overlapping a bright ionized gas filament and a spur of excess X-ray emission, suggesting that this is a location of rapid cooling. The molecular gas is unusually extended, out to ∼55 kpc radius, comparable to the scale of the filamentary nebula in the Perseus cluster, despite the much higher redshift of this system. The extent falls within the thermal instability radius of the intracluster medium (ICM), with tcool/$t_{\rm ff} \lt 25$ and tcool/$t_{\rm eddy} \sim 1$ within ∼70 kpc. Continuum measurements at 159.9 GHz from NOEMA and 850 $\mu$m from the James Clerk Maxwell Telescope (JCMT) Submillimeter Common-User Bolometer Array 2 (SCUBA-2) show excess far-infrared emission, which we interpret as free–free emission arising from the ongoing starburst. These observations suggest that ICM cooling is not strongly affected by the buried QSO, and that cooling from the ICM can build gas reservoirs sufficient to fuel quasar-mode activity and drive the reorientation of the central active galactic nuclei (AGN).