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.