Let $U\subsetneq\mathbb R^n$ be open, let $0<\gamma<1$, and let \begin{align*} Lu&=-\sum_{i,j=1}^n a_{ij}\partial_{x_i}\partial_{x_j}u+\sum_{i=1}^n b_i\partial_{x_i}u+c u \end{align*} be uniformly elliptic in nondivergence form with ellipticity constant $\theta>0$. Suppose $a_{ij},b_i,c\in C^{0,\gamma}(U)$ for $1\le i,j\le n$. If $u\in C^{2,\gamma}(U)$ and $Lu=f$ with $f\in C^{0,\gamma}(U)$, then whenever $\overline W\subsetneq V$ and $\overline V$ is compactly contained in $U$, \begin{align*} \|u\|_{C^{2,\gamma}(\overline W)}&\le C\left(\|f\|_{C^{0,\gamma}(\overline V)}+\|u\|_{C^0(\overline V)}\right), \end{align*} where $C$ depends on $n$, $\gamma$, $\theta$, $W$, $V$, and the $C^{0,\gamma}(\overline V)$ norms of the coefficients.