  We report the detection of CO emission in the recently discovered multiphase
isolated gas cloud in the nearby galaxy cluster Abell 1367. The cloud is
located about 800 kpc in projection from the center of the cluster and at a
projected distance of > 80 kpc from any galaxy. It is the first and the only
known isolated intra-cluster cloud detected in X-ray, H$\alpha$, and CO
emission. We found a total of about $2.2\times 10^8 M_\odot$ of H$_2$ with the
IRAM 30-m telescope in two regions, one associated with the peak of H$\alpha$
emission and another with the peak of X-ray emission surrounded by weak
H$\alpha$ filaments. The velocity of the molecular gas is offset from the
underlying H$\alpha$ emission by > 100 km s$^{-1}$ in the region where the
X-ray peaks. The molecular gas may account for about 10% of the total cloud's
mass, which is dominated by the hot X-ray component. The previously measured
upper limit on the star formation rate in the cloud indicates that the
molecular component is in a non-star-forming state, possibly due to a
combination of low density of the gas and the observed level of velocity
dispersion. The presence of the three gas phases associated with the cloud
suggests that gas phase mixing with the surrounding intra-cluster medium is
taking place. The possible origin of the orphan cloud is a late evolutionary
stage of a ram pressure stripping event. In contrast, the nearby ram pressure
stripped galaxy 2MASX J11443212+2006238 is in an early phase of stripping and
we detected about $2.4\times 10^9 M_\odot$ of H$_2$ in its main body.
