We describe a promising new technique—the "stress-and-relax" method—for reconstructing the coronal magnetic field from vector magnetograph observations of the underlying photosphere. The computation uses the vector potential representation of the coronal field. The starting point for the computation is the potential magnetic field whose normal component at the lower boundary of the computational volume matches the observed normal component. The stress-and-relax method consists of alternately stressing and relaxing the computed coronal field. In the stressing phase, the vector potential in the interior of the computational volume is held fixed, while the vector potential at the lower boundary is adjusted so as to bring the computed transverse field at that boundary into closer agreement with the observed transverse field. This adjustment process distorts the magnetic field lines near the base of the coronal field. In the relaxation phase, the vector potential at the lower boundary is held fixed, while the vector potential in the interior of the computational volume is evolved in such a way that the coronal field relaxes to a nearby force-free configuration. The process of alternately stressing and relaxing the computed coronal magnetic field is continued until the transverse field at the lower boundary is as close as possible to the observed transverse field. The stress-and-relax method is numerically robust, comparatively easy to implement, and it provides a systematic framework for utilizing estimates of the measurement errors in the vector magnetograph data.