  We present a measurement of the branching fraction for the decay of the
neutral $B$ meson into the final state $J/\psi\pi^+\pi^-$. The data set
contains approximately 56 million $B\bar{B}$ pairs produced at the
$\Upsilon(4S)$ resonance and recorded with the BaBar detector at the PEP-II
asymmetric-energy $e^+e^-$ storage ring. The result of this analysis is
$\mathcal{B}$($B^0\to J/\psi\pi^+\pi^-$) = (4.6 $\pm$ 0.7 $\pm$ 0.6)$\times
10^{-5}$, where the first error is statistical and the second is systematic. In
addition we measure $\mathcal{B}(B^0\to J/\psi\rho^0) = (1.6 \pm 0.6 \pm 0.4
)\times 10^{-5}$.
