The low-lying level structure of the unbound neutron-rich nucleus $^{13}$Be has been investigated via breakup on a carbon target of secondary beams of $^{14,15}$B at 35 MeV/nucleon. The coincident detection of the beam velocity $^{12}$Be fragments and neutrons permitted the invariant mass of the $^{12}$Be+$n$ and $^{12}$Be+$n$+$n$ systems to be reconstructed. In the case of the breakup of $^{15}$B, a very narrow structure at threshold was observed in the $^{12}$Be+$n$ channel. Contrary to earlier stable beam fragmentation studies which identified this as a strongly interacting $s$-wave virtual state in $^{13}$Be, analysis here of the $^{12}$Be+$n$+$n$ events demonstrated that this was an artifact resulting from the sequential-decay of the $^{14}$Be(2$^+$) state. Single-proton removal from $^{14}$B was found to populate a broad low-lying structure some 0.70 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)$hbaromega$ shell-model calculations, the low-lying structure is concluded to most probably arise from closely spaced J$^pi$=1/2$^+$ and 5/2$^+$ resonances (E$_r$=0.40$pm$0.03 and 0.85$^{+0.15}_{-0.11}$ MeV), whilst the broad higher-lying feature is a second 5/2$^+$ level (E$_r$=2.35$pm$0.14 MeV). Taken in conjunction with earlier studies, it would appear that the lowest 1/2$^+$ and 1/2$^-$ levels lie relatively close together below 1 MeV.
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