  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 $\eta_{B-L}$ 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 $\eta_{B-L}$ with
the neutrino masses $0.06 eV \leq \bar{m}=(m_{1}^{2}+m_{2}^{2}+m_{3}^{2})^{1/2}
\leq 1.0 eV$, where the full possible cases of degenerate neutrinos are
included. We obtain $\eta_{B-L}\gtrsim 3.3 \times 10^{15} GeV$, $5.3 \times
10^{15} GeV$ and $9.5 \times 10^{15} GeV$ for $\bar{m}=0.2 eV$, $0.4 eV$ and
$1.0 eV$ respectively in the SD process, and find the $B-L$ cosmic string has a
very small contribution to the BAU in the QD process.
