We report the first experimental measurements of the nine 1-fold differential cross sections for the $gamma p to pi^+pi^-p$ reaction, obtained with the CLAS detector at Jefferson Laboratory. The measurements cover the invariant mass range of the final state hadrons from 1.6~GeV~$<W<$~2.0~GeV. For the first time the photocouplings of all prominent nucleon resonances in this mass range have been extracted from this exclusive channel. Photoproduction of two charged pions is of particular importance for the evaluation of the photocouplings for the $Delta(1620)1/2^-$, $Delta(1700)3/2^-$, $N(1720)3/2^+$, and $Delta(1905)5/2^+$ resonances, which have dominant decays into the $pipi N$ final states rather than the more extensively studied single meson decay channels.