An overlapping Schwarz domain decomposition is applied to a semilinear reaction-diffusion two-point boundary value problem with multiple solutions. Its diffusion parameter $\varepsilon^2$ is arbitrarily small, which induces boundary layers. The Schwarz method invokes two boundary-layer subdomains and an interior subdomain, the narrow overlapping regions being of width $O(\varepsilon| \ln \varepsilon|)$. Constructing sub- and super-solutions, we prove existence and investigate the accuracy of discrete solutions in particular subdomains. It is shown that when $\varepsilon \leq CN^{-1}$ and layer-adapted meshes of Bakhvalov and Shishkin types are used, one iteration is sufficient to get second-order convergence (with, in the case of the Shishkin mesh, a logarithmic factor) in the maximum norm uniformly in $\varepsilon$, where $N$ is the number of mesh intervals in each subdomain. Numerical results are presented to support our theoretical conclusions.