  Chlorine and molecular hydrogen are known to be tightly linked together in
the cold phase of the local interstellar medium through rapid chemical
reactions. We present here the first systematic study of this relation at high
redshifts using H$_2$-bearing damped Ly$\alpha$ systems (DLAs) detected along
quasar lines of sight. Using high-resolution spectroscopic data from VLT/UVES
and Keck/HIRES, we report the detection of Cl$\,$I in 9 DLAs (including 5 new
detections) out of 18 high-$z$ DLAs with $N($H$_2) \ge 10^{17.3}\,$cm$^{-2}$
(including a new H$_2$ detection at $z=3.09145$ towards J$\,$2100$-$0641) and
present upper limits for the remaining 9 systems. We find a $\sim$5$\,\sigma$
correlation between $N$(Cl$\,$I) and $N$(H$_2$) with only $\sim$0.2$\,$dex
dispersion over the range 18.1$\,<\,$log$\,N$(H$_2$)$\,<\,$20.1, thus probing
column densities 10 times lower those seen towards nearby stars, roughly
following the relation $N$(Cl$\,$I$) \approx 1.5\times10^{-6} \times N($H$_2)$.
This relation between column densities is surprisingly the same at low and high
redshift suggesting that the physical and chemical conditions are similar for a
given H$_2$ (or Cl$\,$I) column density. In turn, the $N({Cl$\,$I})/N({\rm
H_2})$ ratio is found to be uncorrelated with the overall metallicity in the
DLA. Our results confirm that neutral chlorine is an excellent tracer of
molecule-rich gas and show that the molecular fraction or/and metallicity in
the H$_2$-bearing component of DLA could possibly be much higher than the
line-of-sight average values usually measured in DLAs.
