The Poisson problem consists in finding an immersed surface Σ ⊂ R m minimising Germain's elastic energy (known as Willmore energy in geometry) with prescribed boundary, boundary Gauss map and area. This problem represents a non-linear model for the equilibrium state of thin, clamped elastic plates originating from the work of S. Germain and S.D. Poisson in the early XIX century. We present a solution to this problem in the case of boundary data of class C 1,1 and when the boundary curve is simple and closed. The minimum is realised by an immersed disk, possibly with a finite number of branch points in its interior, which is of class C 1,α up to the boundary for some 0 < α < 1, and whose Gauss map extends to a map of class C 0,α up to the boundary.