  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 38 MeV. This reaction gives maximal possible weight to the
second Lambda(1405) state found in chiral unitary theories, which is narrow and
of higher energy than the nominal Lambda(1405). In contrast, the pi^- p --> K^0
pi Sigma reaction 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 for the first time a clear experimental evidence
of the two pole structure of the Lambda(1405).
