This article is concerned with the adaptive stabilising control design for the ordinary differential equation systems with uncertain diffusion-dominated actuator dynamics. This problem is highly difficult to solve and worthy of investigation, mainly due to the unknown diffusion coefficient which does not belong to any known finite interval and essentially different from the existing literature. By introducing an infinite-dimensional backstepping transformation, the pivotal target system is thus obtained, which makes the control design and stability analysis become more convenient. Then, based on the idea of certainty equivalence principle and recently developed adaptive technique, an adaptive stabilising controller is successfully constructed, which guarantees that all the closed-loop system states are bounded while the original system states converge to zero. A simulation example is presented to illustrate the effectiveness of the proposed method.