  We study the possibility that a large fraction of the gas at temperatures of
\~10^7 K in cooling flow clusters cools by heat conduction to lower
temperatures, rather than by radiative cooling. We argue that this process,
when incorporated into the so-called "moderate cooling flow model", where the
effective age of the intracluster medium is much lower than the age of the
cluster, reduces substantially the expected X-ray luminosity from gas residing
at temperatures of <10^7 K. In this model, the radiative mass cooling rate of
gas at ~10^7 K inferred from X-ray observations, which is <20 % of the mass
cooling rates cited in the past, is easily met. The heat conduction is
regulated by reconnection between the magnetic field lines in cold (~10^4 K)
clouds and the field lines in the intracluster medium. A narrow conduction
front is formed, which, despite the relatively low temperature, allows
efficient heat conduction from the hot ICM to the cold clouds. The reconnection
between the field lines in cold clouds and those in the intracluster medium
occurs only when the magnetic field in the ICM is strong enough. This occurs
only in the very inner regions of cooling flow clusters, at r~10-30 kpc. The
large ratio of the number of H\alpha photons to the number of cooling hydrogen
atoms is explained by this scenario.
