In the early Universe, as a consequence of U (1) B−L gauge symmetry-breaking, the so-called B−L cosmic strings are expected to be produced at the breaking scale η B−L according to the Kibble mechanism. The decaying, collapsing closed loops of these strings can release the right-handed neutrinos, whose subsequent decay can contribute to the baryon asymmetry of the Universe (BAU), through the "slow death"(SD) process and/or the "quick death"(QD) process. In this paper, we assume that the decay of the lightest heavy Majorana neutrinos released from the B − L cosmic string loops can produce a baryon asymmetry consistent with the cosmic microwave background (CMB) observations. Considering the fact that both the neutrinoless double beta decay experiment and the cosmological data show a preference for degenerate neutrinos, we give the lower limits for the breaking scale η B−L with the neutrino masses 0.06eV ≤m = (m 2 1 +m 2 2 +m 2 3 ) 1/2 ≤ 1.0eV, where the full possible cases of degenerate neutrinos are included. We obtain η B−L > ∼ 3.3 × 10 15 GeV, 5.3 × 10 15 GeV and 9.5 × 10 15 GeV form = 0.2eV, 0.4eV and 1.0eV respectively in the SD process, and find the B − L cosmic string has a very small contribution to the BAU in the QD process.