The decay ¯¯¯B0s→J/ψπ+π− can be exploited to study CP violation. A detailed understanding of its structure is imperative in order to optimize its usefulness. An analysis of this three-body final state is performed using a 1.0  fb−1 sample of data produced in 7 TeV pp collisions at the LHC and collected by the LHCb experiment. A modified Dalitz plot analysis of the final state is performed using both the invariant mass spectra and the decay angular distributions. The π+π− system is shown to be dominantly in an S-wave state, and the CP-odd fraction in this ¯¯¯B0s decay is shown to be greater than 0.977 at 95% confidence level. In addition, we report the first measurement of the J/ψπ+π− branching fraction relative to J/ψϕ of (19.79±0.47±0.52)%.