We study for the first time the pΣ − → K − d and K − d → pΣ − reactions close to threshold and show that they are driven by a triangle mechanism, with the Λ(1405), a proton and a neutron as intermediate states, which develops a triangle singularity close to the Kd threshold. We find that a mechanism involving virtual pion exchange and the K − p → π + Σ − amplitude dominates over another one involving kaon exchange and the K − p → K − p amplitude. Moreover, of the two Λ( 1405) states, the one with higher mass around 1420 MeV, gives the largest contribution to the process. We show that the cross section, well within measurable range, is very sensitive to different models that, while reproducing KN observables above threshold, provide different extrapolations of the KN amplitudes below threshold. The observables of this reaction will provide new constraints on the theoretical models, leading to more reliable extrapolations of the KN amplitudes below threshold and to more accurate predictions of the Λ(1405) state of lower mass. A 2 3 1 B C FIG. 1. Feynman diagram from where a triangle singularity can emerge. Particle A decays into 1 and 2, 1 decays into B and 3, and 2 and 3 merge to form particle C.