We present results for a time-dependent Dalitz plot measurement of C P -violating asymmetries in the mode B 0 → π + π − π 0 . The data set is derived from the complete sample of 471 × 10 6 B ¯¯¯ B meson pairs collected with the BABAR detector at the PEP-II asymmetric-energy e + e − collider at the SLAC National Accelerator Laboratory operating on the Υ ( 4 S ) resonance. We extract parameters describing the time-dependent B 0 → ρ π decay probabilities and C P asymmetries, including C = 0.016 ± 0.059 ± 0.036 , Δ C = 0.234 ± 0.061 ± 0.048 , S = 0.053 ± 0.081 ± 0.034 , and Δ S = 0.054 ± 0.082 ± 0.039 , where the uncertainties are statistical and systematic, respectively. We perform a two-dimensional likelihood scan of the direct C P -violation asymmetry parameters for B 0 → ρ ± π ∓ decays, finding the change in χ 2 between the minimum and the origin (corresponding to no direct C P violation) to be Δ χ 2 = 6.42 . We present information on the C P -violating parameter α in a likelihood scan that incorporates B ± → ρ π measurements. To aid in the interpretation of our results, statistical robustness studies are performed to assess the reliability with which the true values of the physics parameters can be extracted. Significantly, these studies indicate that α cannot be reliably extracted with our current sample size, though the other physics parameters are robustly extracted.