The degenerate Neumann problem
\begin{cases}
\ \displaystyle \sum_{i,j=1}^{n}a^{ij}(x)\frac{\partial^{2}u}{\partial x_i\partial x_j}=f(x,u,Du) & \text{in}\ \Omega ,\\
\ a(x)\dfrac{\partial u}{\partial v}+b(x)u=\varphi(x) & \text{on}\ \Gamma
\end{cases}
is studied in the case where a(x) and b(x) are non-negative functions on \Gamma such that a(x)+b(x)>0 on \Gamma.
A classical existence and uniqueness theorem in the Hölder space C^{2+\alpha}(\bar{\Omega}) is proved under suitable regularity and structure conditions on the data.