In this paper, we establish a Hölder continuity with loss of order one for the Cauchy-Riemann equation on a class of smoothly bounded, convex domains of infinite type in the sense of Range in $\mathbb {C}^{3}$. Let $\Omega$ be such a domain and let $\varphi$ be a $(0,1)$-form defined continuously on $\bar {\Omega }$. Then, if $\varphi$ is Lipschitz continuity on $b\Omega$, in the sense of distributions, there exists a function $u$ belonging to a “suitable” Hölder class such that $$\bar{\partial} u=\varphi \quad \text{ in } {\Omega}.$$