We introduce a direct numerical treatment of nonlinear higher-index differential-algebraic equations by means of overdetermined polynomial least-squares collocation. The procedure is not much more computationally expensive than standard collocation methods for regular ordinary differential equations. The numerical experiments show impressive results. In contrast, the theoretical basic concept turns out to be considerably challenging. So far, quite recently convergence proofs for linear problems have been published. In the present paper we come up to a first convergence result for nonlinear problems. (Michael Hanke), maerz@math.hu-berlin.de (Roswitha März) 1 The number of unknowns equals the number of equations. 2 More precisely: Essentially ill-posed in Tichonov's sense, that is, the related operators feature nonclosed ranges.