Conventional vector-controlled voltage-source converters (VSC) suffer from stability and performance problems when integrated into very weak ac grids (high-impedance grids). This is attributed to the increased coupling between the converter and grid dynamics, via the phase-locked loop (PLL), at the very weak-grid condition. In the current literature, the impact of the PLL is usually ignored when the converter controllers are designed. While this approach can be accepted under stiff grid conditions, it yields unmodeled dynamics that destabilize the converter under weak-grid conditions and high-power injection levels. To overcome this limitation, this paper presents a robust vector-controlled VSC that facilitates full converter power injection at weak and very weak-grid conditions (i.e., when the short-circuit capacity ratio is one). To realize the controller, first, a detailed dynamic model for the ac-bus voltage dynamics, including the PLL dynamics, is developed and validated in this paper. Second, the model is used to optimally design a robust ac-bus voltage controller to stabilize the dynamics under operating point variation and grid impedance uncertainty. Because the developed model includes the PLL dynamics, the developed controller inherently stabilizes the negative impact of the PLL on the converter stability. A theoretical analysis and comparative simulation and experimental results are provided to show the effectiveness of the proposed controller.