We present a measurement of the branching fraction for the decay of the neutral B meson into the final state J/psipi(+)pi(-). The data set contains approximately 56 x 10(6) BB 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 B(B0-->J/psipi(+)pi(-))=(4.6+/-0.7+/-0.6) x 10(-5), where the first error is statistical and the second is systematic. In addition, we measure B(B0-->J/psirho(0))=(1.6+/-0.6+/-0.4) x 10(-5).