This thesis deals with the hyper-singular integral equation on a two-dimensional Lipschitz domain with piecewise smooth boundary. We employ a Galerkin method to approximate the solution by non-uniform rational B-splines (NURBS). For this discretization, we derive a weighted residual error estimator with an oscillation term, which stems from the approximation of the given boundary data. We show reliability and other structural properties of this error estimator. Moreover, we employ the error estimator to steer an adaptive algorithm. The main results read as follows: First, the error estimator is linearly convergent with respect to the number of adaptive steps. Second, it decays at optimal algebraic convergence rate. Numerical results underpin our theoretical results. The appendix provides the Matlac/C-code of our implementation.