The interaction of an E/A=57.6-MeV ^{17}Ne beam with a Be target is used to populate levels in ^{16}Ne following neutron knockout reactions. The decay of ^{16}Ne states into the three-body ^{14}O+p+p continuum is observed in the High Resolution Array (HiRA). For the first time for a 2p emitter, correlations between the momenta of the three decay products are measured with sufficient resolution and statistics to allow for an unambiguous demonstration of their dependence on the long-range nature of the Coulomb interaction. Contrary to previous measurements, our measured limit Γ<80 keV for the intrinsic decay width of the ground state is not in contradiction to the small values (of the order of keV) predicted theoretically.