  We consider the ortho-- and parapositronium production in the process $AA \to
AA+$ Ps where A is a nucleus with the charge number Z. The inclusive cross
section and the energy distribution of the relativistic Ps are calculated which
are of primary interest from the experimental point of view. The accuracy of
the corresponding cross sections is given by omitting terms $\sim (Z\alpha
)^2/L^2$ for the para--Ps and $\sim (Z\alpha)^2/L$ for the ortho--Ps production
where $L=\ln{\gamma^2} \approx 9$ and 16 for the RHIC and the LHC. Within this
accuracy the multiphoton (Coulomb) corrections are taken into account. We show
that the RHIC and the LHC will be Ps factories with a productions rate of about
$10^5 \div 10^8$ relativistic Ps per day. The fraction of the ortho--Ps is
expected to be of the same order as that of the para--Ps for Au--Au and Pb--Pb
collisions.
