Let \(\Omega\) be an open subset of \(\mathbb R^n\), let \(\mathcal{V}\) be a closed subspace of \(H^1(\Omega)\), and \(J: \mathcal{V} \times \mathcal{V} \to \mathbf C\) a sesquilinear form, such that, for some \(\kappa > 0\), \(\operatorname{Re} J[u,u] \geq \kappa \|u\|_{H^1(\Omega)}^2\), for all \(u \in \mathcal{V}\). It is well known that there exists a unique maximal accretive operator \(L : D(L) \subseteq \mathcal{V} \to L^2(\Omega)\), such that \(J[u,v] = (Lu,v)_{L^2(\Omega)}\), \(\forall u \in D(L), \forall v \in \mathcal{V}\). The Kato square root problem consists in establishing whether the domain of the square root \(\sqrt{L}\) coincides with \(\mathcal{V}\). The authors give a positive answer in the case of a form coming from an second elliptic operator in variational form, with mixed (Neumann-Dirichlet) boundary conditions. Contrary to previous papers, the coefficients of the principal part are assumed only in \(L^\infty(\Omega)\). Techniques developed in this paper build upon ideas introduced in a paper due to the same authors [see Invent. Math. 163, 455–497 (2006; Zbl 1094.47045)].