  We have performed a detailed investigation of total lifetimes for the
$\Xi_{cc}^{++}$ and $\Xi_{cc}^{+}$ baryons in the framework of operator product
expansion over the inverse mass of charmed quark, whereas, to estimate matrix
elements of operators obtained in OPE, some approximations of nonrelativistic
QCD are used. This approach allows one to take into account the corrections to
the spectator decays of $c$-quarks, which reflect the fact, that these quarks
are bound, as well as the contributions, connected to the effects of both the
Pauli interference for the $\Xi_{cc}^{++}$-baryon and the weak scattering for
the $\Xi_{cc}^{+}$-baryon. The realization of such program leads to the
following estimates for the total lifetimes of doubly charmed baryons:
$\tau_{\Xi_{cc}^{++}}=0.43\pm 0.1~ps$ and $\tau_{\Xi_{cc}^{+}}=0.11\pm
0.01~ps$.
