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We consider Walsh's conformal map from the exterior of a set$E=\bigcup_{j=1}^\ell E_j$ consisting of $\ell$ compact disjoint componentsonto a lemniscatic domain. In particular, we are interested in the case when$E$ is a polynomial preimage of $[-1,1]$, i.e., when $E=P^{-1}([-1,1])$, where$P$ is an algebraic polynomial of degree $n$. Of special interest are theexponents and the centers of the lemniscatic domain. In the first part of thisseries of papers, a very simple formula for the exponents has been derived. Inthis paper, based on general results of the first part, we give an iterativemethod for computing the centers when $E$ is the union of $\ell$ intervals.Once the centers are known, the corresponding Walsh map can be computednumerically. In addition, if $E$ consists of $\ell=2$ or $\ell=3$ componentssatisfying certain symmetry relations then the centers and the correspondingWalsh map are given by explicit formulas. All our theorems are illustrated withanalytical or numerical examples.

Walsh's Conformal Map onto Lemniscatic Domains for Polynomial Pre-images II