Using (1.0087±0.0044)×10^{10} J/ψ events collected with the BESIII detector at the BEPCII storage ring, the process Ξ^{0}n→Ξ^{-}p is studied, where the Ξ^{0} baryon is produced in the process J/ψ→Ξ^{0}Ξ[over ¯]^{0} and the neutron is a component of the ^{9}Be, ^{12}C, and ^{197}Au nuclei in the beam pipe. A clear signal is observed with a statistical significance of 7.1σ. The cross section of the reaction Ξ^{0}+^{9}Be→Ξ^{-}+p+^{8}Be is determined to be σ(Ξ^{0}+^{9}Be→Ξ^{-}+p+^{8}Be)=(22.1±5.3_{stat}±4.5_{sys}) mb at the Ξ^{0} momentum of 0.818 GeV/c, where the first uncertainty is statistical and the second is systematic. No significant H-dibaryon signal is observed in the Ξ^{-}p final state. This is the first study of hyperon-nucleon interactions in electron-positron collisions and opens up a new direction for such research.