  We prove that if $X$ and $S$ are smooth varieties and $f\colon X\to S$ is an
elliptic fibration with singular fibers curves of types I$_N$ with $N\geq 1$,
II, III and IV, then the relative Jacobian $\hat{f}\colon \bar{M}_{X/S}\to S$
of $f$, defined as the relative moduli space of semistable pure dimension one
sheaves of rank 1 and degree 0 on the fibers of $f$, is an elliptic fibration
such that all its fibers are irreducible. This extends known results when
fibers are integral or of type I$_2$.
