We report on precision measurements of the masses and widths of the narrow, orbitally excited states D01 and D*02 using the CDF II detector at the Fermilab Tevatron. Both states (collectively called D**) are reconstructed in the decay channel D**→D*+π−. The D*02 is also reconstructed in the D**→D+π− channel. Using a data set with an integrated luminosity of 210  pb−1, the measured masses and widths for the D01 are 2421.7±0.7±0.6   MeV/c2 and 20.0±1.7±1.3   MeV/c2 respectively, while for the D*02 they are 2463.3±0.6±0.8   MeV/c2 and 49.2±2.3±1.2   MeV/c2. These values are currently the single best measurements available.