  We have imaged in CO(2-1) the molecular gas in NGC 1275 (Perseus A), the cD
galaxy at the center of the Perseus Cluster, at a spatial resolution of $\sim$1
kpc over a central region of radius $\sim$ 10 kpc. Per A is known to contain
$\sim$1.3x10$^{10}$ M$_\odot$ of molecular gas, which has been proposed to be
captured from mergers with or ram-pressure stripping of gas-rich galaxies, or
accreted from a X-ray cooling flow. The molecular gas detected in our image has
a total mass of $\sim$4x10$^9$ M$_\odot$, and for the first time can be seen to
be concentrated in three radial filaments with lengths ranging from at least
1.1-2.4 kpc all lying in the east-west directions spanning the center of the
galaxy to radii of $\sim$8 kpc. The eastern and outer western filaments exhibit
larger blueshifted velocities with decreasing radii, whereas the inner western
filament spans the systemic velocity of the galaxy. The molecular gas shows no
signature of orbital motion, and is therefore unlikely to have been captured
from gas-rich galaxies. Instead, we are able to reproduce the observed
kinematics of the two outer filaments as free-fall in the gravitational
potential of Per A, as would be expected if they originate from a X-ray cooling
flow. Indeed, all three filaments lie between two prominent X-ray cavities
carved out by radio jets from Per A, and closely resembles the spatial
distribution of the coolest X-ray gas in the cluster core. The inferred
mass-deposition rate into the two outermost filaments alone is roughly 75
M$_odot$ yr$^{-1}$. This cooling flow can provide a nearly continuous supply of
molecular gas to fuel the active nucleus in Per A.
