  We present the results of interferometric observations of the cool core of
Abell 1795 at CO(1-0) using the Combined Array for Research in Millimeter-Wave
Astronomy. In agreement with previous work, we detect a significant amount of
cold molecular gas (3.9 +/- 0.4 x10^9 Msun) in the central ~10 kpc. We report
the discovery of a substantial clump of cold molecular gas at clustercentric
radius of 30 kpc (2.9 +/- 0.4 x10^9 Msun), coincident in both position and
velocity with the warm, ionized filaments. We also place an upper limit on the
H_2 mass at the outer edge of the star-forming filament, corresponding to a
distance of 60 kpc (<0.9 x10^9 Msun). We measure a strong gradient in the
HII/H_2 ratio as a function of radius, suggesting different ionization
mechanisms in the nucleus and filaments of Abell1795. The total mass of cold
molecular gas (\sim7x10^9 Msun) is roughly 30% of the classical cooling
estimate at the same position, assuming a cooling time of 10^9 yr. Combining
the cold molecular gas mass with the UV-derived star formation rate and the
warm, ionized gas mass, the spectroscopically-derived X-ray cooling rate is
fully accounted for and in good agreement with the cooling byproducts over
timescales of \sim10^9 yr. The overall agreement between the cooling rate of
the hot intracluster medium and the mass of the cool gas reservoir suggests
that, at least in this system, the cooling flow problem stems from a lack of
observable cooling in the more diffuse regions at large radii.
