Résumé : The low-lying level structure of the unbound neutron-rich nucleus 13Be has been investigated via breakup on a carbon target of secondary beams of 14,15B at 35 MeV/nucleon. The coincident detection of the beam velocity 12Be fragments and neutrons permitted the invariant mass of the 12Be+n and 12Be+n+n systems to be reconstructed. In the case of the breakup of 15B, a very narrow structure at threshold was observed in the 12Be+n channel. Analysis of the 12Be+n+n events demonstrated that this resulted from the sequential decay of the unbound 14Be(2+) state rather than a strongly interacting s-wave virtual state in 13Be, as had been surmised in stable beam fragmentation studies. Single-proton removal from 14B was found to populate a broad low-lying structure some 0.7 MeV above the neutron-decay threshold, in addition to a less prominent feature at around 2.4 MeV. Based on the selectivity of the reaction and a comparison with (0-3) ω shell-model calculations, the low-lying structure is concluded to arise from closely spaced Jπ=1/2+ and 5/2+ resonances (Er=0.40±0.03 and 0.85-0.11+0.15 MeV), while the broad higher-lying feature is a second 5/2+ level (Er=2.35±0.14 MeV). Taken in conjunction with earlier studies, the results suggest that the lowest 1/2+ and 1/2- levels lie relatively close together below 1 MeV. © 2014 American Physical Society.