  We characterize the importance of metallicity on the presence of molecular
hydrogen in damped Lyman-alpha (DLA) systems. We construct a representative
sample of 18 DLA/sub-DLA systems with log N(HI)>19.5 at high redshift
(zabs>1.8) with metallicities relative to solar [X/H]>-1.3(with[X/H]=
logN(X)/N(H)-log(X/H)solar and X either Zn, S or Si). We gather data covering
the expected wavelength range of redshifted H2 absorption lines on all systems
in the sample from either the literature (10 DLAs), the UVES-archive or new
VLT-UVES observations for four of them. The sample is large enough to discuss
for the first time the importance of metallicity as a criterion for the
presence of molecular hydrogen in the neutral phase at high-z. From the new
observations, we report two new detections of molecular hydrogen in the systems
at zabs=2.431 toward Q2343+125 and zabs=2.426 toward Q2348-011. We compare the
H2 detection fraction in the high-metallicity sample with the detection
fraction in the overall sample from Ledoux et al. (2003). We show that the
fraction of DLA systems with logf=log 2N(H2)/(2N(H2)+N(HI))>-4 is as large as
50% for [X/H]>-0.7 when it is only about 5% for [X/H]<-1.3 and about 15% in the
overall sample (with -2.5<[X/H]<-0.3). This demonstrates that the presence of
molecular hydrogen at high redshift is strongly correlated with metallicity.
