We study the decays B0→J/ψπ+π- and B+→J/ ψπ+π0, including intermediate resonances, using a sample of 382×106 BB̄ pairs recorded by the BABAR detector at the PEP-II e+e- B factory. We measure the branching fractions B(B0→J/ψρ0)=(2. 7±0.3±0.2)×10-5 and B(B+→J/ψρ+)=(5.0±0. 7±0.3)×10-5. We also set the following upper limits at the 90% confidence level: B(B0→J/ψπ+π-nonresonant)<1.2×10-5, B(B0→J/ψf2)<4.6×10-6, and B(B+→J/ ψπ+π0nonresonant)<7.3×10-6. We measure the charge asymmetry in charged B decays to J/ψρ to be -0.11±0.12±0.08. © 2007 The American Physical Society.