Let S subset of R-2 be a bounded domain with boundary of class C-infinity, and let g(ij) = delta(ij) denote the flat metric on R-2. Let u be a minimizer of the Willmore functional within a subclass (defined by prescribing boundary conditions on parts of partial derivative S) of all W-2,W-2 isometric immersions of the Riemannian manifold. (S, g) into R-3. In this article we derive the Euler-Lagrange equation and study the regularity properties for such u. Our main regularity result is that minimizers u are C-3 away from a certain singular set Sigma and C-infinity away from a larger singular set Sigma boolean OR Sigma(0). We obtain a geometric characterization of these singular sets, and we derive the scaling of u and its derivatives near Sigma(0). Our main motivation to study this problem comes from nonlinear elasticity: On isometric immersions, the Willmore functional agrees with Kirchhoff's energy functional for thin elastic plates.