  We discuss the possibility of extracting ${\mid}V_{cb}{\mid}$ from the future
measurement at LHC of the $B_c\ {\rightarrow}\ J/\psi\ {\mu}^+\ {\nu}$ decay
rate followed by the leptonic decay $J/\psi\ {\rightarrow}\ \mu^+\mu^-$. We
examine in detail the influence of the measurement of the muons' momenta and
vertex reconstruction in the final uncertainty of ${\mid}V_{cb}{\mid}$, as well
as the required theoretical inputs at both the production and weak decay level,
and their reliability in this regard. We stress the relevance of a rigorous
description of the fragmentation contributions to inclusive production at large
transverse momentum of heavy quarkonia in $pp$ collisions.
