  In the type-II seesaw mechanism, the neutrino mass generation could be tested
experimentally if the Higgs triplet is at the TeV scale and has a small cubic
coupling to the standard model Higgs doublet. We show such small
triplet-doublet coupling and the cosmic baryon asymmetry can be simultaneously
induced by an additional seesaw mechanism involving a $U(1)_{B-L}$ gauge
symmetry. Meanwhile, three neutral fermions for cancelling the gauge anomalies
can form a stable Dirac fermionic dark matter besides an acceptably massless
fermion.
