This paper addresses the problem of dynamic frequency control in a diesel-based mini-grid. It is shown that a virtual synchronous machine (VSM) can support dynamic frequency control by adding virtual inertia and damping to the system. However, it is found that the typical formulation of damping power does not work properly when the grid forming gen-set operates in droop mode because of the unknown stabilization value of the grid frequency. As a solution to this problem, an estimator for the stabilization frequency that works in conjunction with the damping function of the VSM is proposed. Theoretical and experimental results provide evidence of a satisfactory performance of the proposed VSM with estimator for different values of the gen-set droop factor. The estimated stabilization frequency converges in approximately 2 s and the maximum frequency deviation during the transient is reduced in 34%, on average.