A domain decomposition procedure for solving parabolic partial differential equations is presented. In this scheme, a Galerkin finite element discretization on a rectangular mesh is used. Boundary values at subdomain interfaces are calculated by an implicit/explicit procedure. This solution serves as boundary data for fully implicit subdomain problems, which can be solved simultaneously. Thus, the scheme is non-iterative, and requires non-overlapping subdomains. Numerical results are presented for problems in two space dimensions.