Active galactic nuclei play a crucial role in the accretion and ejection of gas in galaxies. Although their outflows are well studied, finding direct evidence of accretion has proved very difficult and has so far been done for very few sources. A promising way to study the significance of cold accretion is by observing the absorption of an active galactic nucleuss extremely bright radio emission by the cold gas lying along the line-of-sight. As such, we present ALMA CO(1-0) and CO(2-1) observations of the Hydra-A brightest cluster galaxy (z=0.054) which reveal the existence of cold, molecular gas clouds along the line-of-sight to the galaxys extremely bright and compact mm-continuum source. They have apparent motions relative to the central supermassive black hole of between -43 and -4 km s$^{-1}$ and are most likely moving along stable, low ellipticity orbits. The identified clouds form part of a $sim$$10^{9}$ $text{M}_{odot}$, approximately edge-on disc of cold molecular gas. With peak CO(2-1) optical depths of $tau$=0.88 $^{+0.06}_{-0.06}$, they include the narrowest and by far the deepest absorption of this type which has been observed to date in a brightest cluster galaxy. By comparing the relative strengths of the lines for the most strongly absorbing region, we are able to estimate a gas temperature of $42^{+25}_{-11}$ K and line-of-sight column densities of $N_{CO}=2^{+3}_{-1}times 10 ^{17} cm^{-2}$ and $N_{ H_{2} }=7^{+10}_{-4}times 10 ^{20} cm^{-2}$.
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