We present the direct detection of molecular hydrogen at the highest redshift
known today (z_abs=4.224) in a Damped Lyman-alpha (DLA) system toward the
quasar PSS J1443+2724. This absorber is remarkable for having one of the
highest metallicities amongst DLA systems at z_abs>3, with a measured iron
abundance relative to Solar of -1.12+/-0.10. We provide for the first time in
this system accurate measurements of NI, MgII, SII and ArI column densities.
The sulfur and nitrogen abundances relative to Solar, -0.63+/-0.10 and
-1.38+/-0.10 respectively, correspond exactly to the primary nitrogen
production plateau. H2 absorption lines are detected in four different
rotational levels (J=0, 1, 2 and 3) of the vibrational ground-state in three
velocity components with total column densities of log N(H2)=17.67, 17.97,
17.48 and 17.26 respectively. The J=4 level is tentatively detected in the
strongest component with log N(H2)~14. The mean molecular fraction is log
f=-2.38+/-0.13, with f=2N(H2)/(2N(H2)+N(HI)). We also measure log
N(HD)/N(H2)<-4.2. The excitation temperatures T_{01} for the two main
components of the system are 96 and 136 K respectively. We argue that the
absorbing galaxy, whose star-formation activity must have started at least
2-5x10^8 yrs before z=4.224, is in a quiescent state at the time of
observation. The density of the gas is small, n_H<=50 cm^{-3}, and the
temperature is of the order of T~90-180 K. The high excitation of neutral
carbon in one of the components can be explained if the temperature of the
Cosmic Microwave Background Radiation has the value expected at the absorber
redshift, T=14.2 K.Comment: 13 pages, 3 figures, accepted for publication in ApJ Letter