  The dressing chain is derived by applying Darboux transformations to the
spectral problem of the Korteweg-de Vries (KdV) equation. It is also an
auto-B\"acklund transformation for the modified KdV equation. We show that by
applying Darboux transformations to the spectral problem of the dressing chain
one obtains the lattice KdV equation as the dressing chain of the dressing
chain and, that the lattice KdV equation also arises as an auto-B\"acklund
transformation for a modified dressing chain. In analogy to the results
obtained for the dressing chain (Veselov and Shabat proved complete
integrability for odd dimensional periodic reductions), we study the
$(0,n)$-periodic reduction of the lattice KdV equation, which is a two-valued
correspondence. We provide explicit formulas for its branches and establish
complete integrability for odd $n$.
