  In this paper, we prove a $\mathcal C^{2,\alpha}$-estimate for the solution
to the complex Monge-Amp\`ere equation $\det(u_{i\bar{j}})=f$ with $0< f\in
\mathcal C^{\alpha}$, under the assumption that $u\in \mathcal C^{1,\beta }$
for some $\beta <1$ which depends on $n$ and $\alpha $.
