  We study the effect on leptogenesis due to $B-L$ cosmic strings of a
$U(1)_{B-L}$ extension of the Standard Model. The disappearance of closed loops
of $B-L$ cosmic strings can produce heavy right handed neutrinos, $N_R$'s,
whose CP-asymmetric decay in out-of-thermal equilibrium condition can give rise
to a net lepton ($L$) asymmetry which is then converted, due to sphaleron
transitions, to a Baryon ($B$) asymmetry. This is studied by using the relevant
Boltzmann equations and including the effects of both thermal and string
generated non-thermal $N_R$'s. We explore the parameter region spanned by the
effective light neutrino mass parameter $\tilde{m}_1$, the mass $M_1$ of the
lightest of the heavy right-handed neutrinos (or equivalently the Yukawa
coupling $h_1$) and the scale of $B-L$ symmetry breaking, $\eta_{B-L}$, and
show that there exist ranges of values of these parameters, in particular with
$\eta_{B-L} > 10^{11}\gev$ and $h_1\gsim 0.01$, for which the cosmic string
generated non-thermal $N_R$'s can give the dominant contribution to, and indeed
produce, the observed Baryon Asymmetry of the Universe when the purely thermal
leptogenesis mechanism is not sufficient. We also discuss how, depending on the
values of $\eta_{B-L}$, $\tilde{m}_1$ and $h_1$, our results lead to upper
bounds on $\sin\delta$, where $\delta$ is the the CP violating phase that
determines the CP asymmetry in the decay of the heavy right handed neutrino
responsible for generating the $L$-asymmetry.
