Differential cross sections for the gamma p -> pi^0 p reaction have been measured with the A2 tagged-photon facilities at the Mainz Microtron, MAMI C, up to the center-of-mass energy W=1.9 GeV. The new results, obtained with a fine energy and angular binning, increase the existing quantity of pi^0 photoproduction data by ~47%. Owing to the unprecedented statistical accuracy and the full angular coverage, the results are sensitive to high partial-wave amplitudes. This is demonstrated by the decomposition of the differential cross sections in terms of Legendre polynomials and by further comparison to model predictions. A new solution of the SAID partial-wave analysis obtained after adding the new data into the fit is presented.