  We present a theoretical study of a double lepton pair production in
ultra--relativistic collision between two bare ions. Special emphasis is placed
to processes in which creation of (at least one) $e^+ e^-$ pair is accompanied
by the capture of an electron into a bound ionic state. To evaluate the
probability and cross section of these processes we employ two approaches based
on (i) the first--order perturbation theory and multipole expansion of Dirac
wavefunctions, and (ii) the equivalent photon approximation. With the help of
such approaches, detailed calculations are made for the creation of two
bound--free $e^+ e^-$ pairs as well as of bound--free $e^+ e^-$ and free--free
$\mu^+ \mu^-$ pairs in collisions of bare lead ions Pb$^{82+}$. The results of
the calculations indicate that observation of the double lepton processes may
become feasible at the LHC facility.
