In an emulsion-counter hybrid experiment performed at J-PARC, a Ξ^{-} absorption event was observed which decayed into twin single-Λ hypernuclei. Kinematic calculations enabled a unique identification of the reaction process as Ξ^{-}+^{14}N→_{Λ}^{10}Be+_{Λ}^{5}He. For the binding energy of the Ξ^{-} hyperon in the Ξ^{-}-^{14}N system a value of 1.27±0.21 MeV was deduced. The energy level of Ξ^{-} is likely a nuclear 1p state which indicates a weak ΞN-ΛΛ coupling.