The K^- p --> pi^0 pi^0 Sigma^0 reaction is studied within a chiral unitary model. The distribution of pi^0 Sigma^0 states forming the Lambda(1405) shows, in agreement with a recent experiment, a peak at 1420 MeV and a relatively narrow width of Gamma = 38 MeV. The mechanism for the reaction is largely dominated by the emission of a pi^0 prior to the K^- p interaction leading to the Lambda(1405). This ensures the coupling of the Lambda(1405) to the K^- p channel, thus maximizing the contribution of the second state found in chiral unitary theories, which is narrow and of higher energy than the nominal Lambda(1405). This is unlike the pi^- p --> K^0 \pi Sigma reaction, which gives more weight to the pole at lower energy and with a larger width. The data of these two experiments, together with the present theoretical analysis, provides a firm evidence of the two pole structure of the Lamb... [Llegir més ...]