We describe a domain decomposition algorithm for use in several variants of the parallel adaptive meshing paradigm of Bank and Holst. This algorithm has low communication, makes extensive use of existing sequential solvers, and exploits in several important ways data generated as part of the adaptive meshing paradigm. We show that for an idealized version of the algorithm, the rate of convergence is independent of both the global problem size N and the number of subdomains p used in the domain decomposition partition. Numerical examples illustrate the effectiveness of the procedure.