A steady state viscoelastic model of deformation at an oblique convergence zone is used to analyze crustal velocities deduced from Global Positioning System (GPS) observations in southern North Island, New Zealand. The model is physically more reasonable than elastic dislocation theory because the tectonic plates have finite elastic thicknesses. In an inversion that makes use of Green's functions derived from finite element calculations, we solve for depth-dependent fault backslip rates. The associated chi-squared goodness of fit parameter depends on the values of the elastic thicknesses of the overriding Australian and subducting Pacific Plates. These thicknesses are systematically varied in order to find the chi-squared minimum. We find that: (1) the plates have coupling coefficient between 0.8 and 1.0 to a depth of about 22 km; (2) elastic dislocation theory appears to adequately fit the observations because the effects of viscoelastic flow are small; (3) viscoelastic results depend on the contrast between the elastic moduli of the plates, (4) the trench normal, rather than the trench parallel component of motion is more diagnostic for choosing between models with different parameters; (5) for the favored model (one with a weak continental crust), the estimated value of the Pacific Plate thickness is 40--60 km. Although the estimates of the plate thickness are not tightly constrained, those deduced from geodetic data tend to be larger than those deduced from geologic data, consistent with the idea that thickness estimates depend on the time scale of the loading process.