  The difference in total widths between the $B_c^+$ and $B^+$ mesons is
measured using 3.0fb$^{-1}$ of data collected by the LHCb experiment in 7 and 8
TeV centre-of-mass energy proton-proton collisions at the LHC. Through the
study of the time evolution of $B_c^+ \rightarrow J/\psi \pi^+$ and
$B^+\rightarrow J/\psi K^+$ decays, the width difference is measured to be
  $$ \Delta\Gamma \equiv \Gamma_{B_c^+} - \Gamma_{B^+} = 4.46 \pm 0.14 \pm
0.07mm^{-1}c,$$ where the first uncertainty is statistical and the second
systematic. The known lifetime of the $B^+$ meson is used to convert this to a
precise measurement of the $B_c^+$ lifetime,
  $$\tau_{B_c^+} = 513.4 \pm 11.0 \pm 5.7fs,$$ where the first uncertainty is
statistical and the second systematic.
