A precision measurement of the differential cross sections dσ/dΩ and the linearly polarized photon asymmetry Σ≡(dσ⊥-dσ∥)/(dσ⊥+dσ∥) for the γp→π0p reaction in the near-threshold region has been performed with a tagged photon beam and almost 4π detector at the Mainz Microtron. The Glasgow-Mainz photon tagging facility along with the Crystal Ball/TAPS multiphoton detector system and a cryogenic liquid hydrogen target were used. These data allowed for a precise determination of the energy dependence of the real parts of the S- and all three P-wave amplitudes for the first time and provide the most stringent test to date of the predictions of chiral perturbation theory and its energy region of agreement with experiment.