New precise results of a measurement of the elastic electron-proton scattering cross section performed at the Mainz Microtron MAMI are presented. About 1400 cross sections were measured with negative four-momentum transfers squared up to Q² = 1 (GeV/c)² with statistical errors below 0.2%. The electric and magnetic form factors of the proton were extracted by fits of a large variety of form factor models directly to the cross sections. The form factors show some features at the scale of the pion cloud. The charge and magnetic radii are determined to be <r²E>½ = 0.879(5)stat(4)syst(2)model(4)group fm and <r²M>½ = 0.777(13)stat(9)syst(5)model(2)group fm.