  The decay $B_c^+\to J/\psi K^+$ is observed for the first time using a data
sample, corresponding to an integrated luminosity of 1.0 ${\,fb}^{-1}$,
collected by the LHCb experiment in $pp$ collisions at a centre-of-mass energy
of $7\mathrm{\,Te\kern -0.1em V}$. A yield of $46\pm12$ events is reported,
with a significance of 5.0 standard deviations. The ratio of the branching
fraction of $B_c^+\to J/\psi K^+$ to that of $B_c^+\to J/\psi\pi^+$ is measured
to be $0.069\pm0.019\pm0.005$, where the first uncertainty is statistical and
the second is systematic.
