We present a Dalitz plot analysis of charmless B± decays to the final state π±π±π using a sample of (465±5) ×106 BB̄ pairs collected by the BABAR experiment at s=10.58GeV. We measure the branching fractions B(B±→π±π±π)= (15.2±0.6±1.2±0.4)×10-6, B(B±→ρ0(770) π±)=(8.1±0.7±1.2-1.1+0.4)×10-6, B(B±→f2(1270)π±)=(1.57±0.42±0.16-0.19+0.53) ×10-6, and B(B±→π±π±πnonresonant)=(5. 3±0.7±0.6-0.5+1.1)×10-6, where the uncertainties are statistical, systematic, and model-dependent, respectively. Measurements of branching fractions for the modes B±→ρ0(1450)π± and B±→f0(1370)π± are also presented. We observe no significant direct CP asymmetries for the above modes, and there is no evidence for the decays B±→f0(980)π±, B±→χc0π ±, or B±→χc2π±. © 2009 The American Physical Society.