  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 Q^2=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 r_E=0.879(5)(stat.)(4)(syst.)(2)(model)(4)(group) fm
and r_M=0.777(13)(stat.)(9)(syst.)(5)(model)(2)(group) fm.
