Using data from the CLEO II detector at CESR, we measure B( τ ± → h ± π 0 ν τ) where h ± refers to either π ± or K ±. We use τ + τ − events in which one tau decays to h ± π 0 ν τ and the other decays into either e ± ν eν τ, μ ± ν μ ν τ, h ± π 0 ν τ, or h ± h + h −( π 0) ν τ. We use three different methods to measure this branching fraction. The combined result is B( τ ± → h ± π 0 ν τ) = 0.2587 ± 0.0012 ± 0.0042, in good agreement with predictions from CVC. This result, in combination with other precision measurements, reduces the significance of the one-prong problem in tau decays. Preliminary results from a fit to the π ± π 0 mass distribution are also presented.