This article investigates how the eigenvalues of the Neumann problem for an elliptic operator depend on the domain in the case when the domains involved are of class C-1. We consider the Laplacian and use results developed previously for the corresponding Lipschitz case. In contrast with the Lipschitz case, however, in the C-1-case we derive an asymptotic formula for the eigenvalues when the domains are of class C-1. Moreover, as an application we consider the case of a C-1-perturbation when the reference domain is of class C-1,C-alpha.