The charge and magnetic form factors, F_{C} and F_{M}, respectively, of ^{3}He are extracted in the kinematic range 25 fm^{-2}≤Q^{2}≤61 fm^{-2} from elastic electron scattering by detecting ^{3}He recoil nuclei and scattered electrons in coincidence with the two High Resolution Spectrometers of the Hall A Facility at Jefferson Lab. The measurements find evidence for the existence of a second diffraction minimum for the magnetic form factor at Q^{2}=49.3 fm^{-2} and for the charge form factor at Q^{2}=62.0 fm^{-2}. Both minima are predicted to exist in the Q^{2} range accessible by this Jefferson Lab experiment. The data are in qualitative agreement with theoretical calculations based on realistic interactions and accurate methods to solve the three-body nuclear problem.