The paper proposes an adaptive variable synthetic inertia strategy to provide frequency and voltage support (ancillary services) in an AC MicroGrid (weak grid) composed of diesel generators and loads. A DC MicroGrid is connected to the AC one by a Voltage Source Converter (VSC). The VSC converter is driven as a Virtual Synchronous Machine (VSM), where the control strategy follows a swing equation such the converter emulates a synchronous machine, including inertial support. A rigorous stability analysis is developed based on Lyapunov technique assuring proper stability conditions for an adaptive inertia designated, such that, frequency stability is improved and power oscillations are reduced. This strategy can be exploited in low inertia systems like MicroGrids or grids with high penetration of renewables. Simulation results illustrate the performance of the proposed control and the system's operation, where a comparison with droop control is presented.