  The first two Hilbert coefficients of a primary ideal play an important role
in commutative algebra and in algebraic geometry. In this paper we give a
complete algebraic structure of the Sally module of integrally closed ideals
$I$ in a Cohen-Macaulay local ring $A$ satisfying the equality
$\mathrm{e}_1(I)=\mathrm{e}_0(I)-\ell_A(A/I)+\ell_A(I^2/QI)+1, $ where $Q$ is a
minimal reduction of $I$, and $\mathrm{e}_0(I)$ and $\mathrm{e}_1(I)$ denote
the first two Hilbert coefficients of $I, $ respectively the multiplicity and
the Chern number of $I$. This almost extremal value of $\mathrm{e}_1(I) $ with
respect classical inequalities holds a complete description of the homological
and the numerical invariants of the associated graded ring. Examples are given.
