  We report the results of a search for the $B \to Y(4260) K, ~Y(4260)\to
J/\psi\pi^+\pi^-$ decays. This study is based on a data sample corresponding to
an integrated luminosity of 711~fb$^{-1}$, collected at the $\Upsilon(4S)$
resonance with the Belle detector at the KEKB asymmetric-energy $e^+ e^-$
collider. We investigate the $J/\psi\pi^+\pi^-$ invariant mass distribution in
the range 4.0 to 4.6 GeV/$c^2$ using both $B^+ \to J/\psi \pi^+\pi^- K^+$ and
$B^0 \to J/\psi \pi^+\pi^- K^0_S$ decays. We find excesses of events above the
background levels, with a significances of 2.1 and 0.9 standard deviations for
charged and neutral $B \to Y(4260) K$ decays, respectively, taking into account
the systematic uncertainties. These correspond to upper limits on the product
of branching fractions, ${\cal B}(B^+ \to Y(4260) K^+) \times {\cal B}(Y(4260)
\to J/\psi \pi^+ \pi^-) <1.4 \times 10^{-5}$ and ${\cal B}(B^0 \to Y(4260) K^0)
\times {\cal B}(Y(4260) \to J/\psi \pi^+ \pi^-) <1.7 \times 10^{-5}$ at the
90\% confidence level.
