We present two new primary mechanisms for the synthesis of the rare nucleus Be9, both triggered by ν-induced production of H3 followed by He4(H3,γ)Li7 in the He shells of core-collapse supernovae. For progenitors of ∼8M⊙, Li7(H3,n0)Be9 occurs during the rapid expansion of the shocked He shell. Alternatively, for ultra-metal-poor progenitors of ∼11-15M⊙, Li7(n,γ)Li8(n,γ)Li9(e-ν̄e)Be9 occurs with neutrons produced by He4(ν̄e,e+n)H3, assuming a hard effective ν̄e spectrum from oscillations (which also leads to heavy element production through rapid neutron capture) and a weak explosion (so the Be9 survives shock passage). We discuss the associated production of Li7 and B11, noting patterns in LiBeB production that might distinguish the new mechanisms from others. © 2013 American Physical Society.