Graphene has received widespread attention due to its unique electronic properties. Much of the research conducted so far has focused on electron mobility, which is determined by scattering from charged impurities and other inhomogeneities. However, another important quantity, the quantum capacitance, has been largely overlooked. Here, we report a direct measurement of the quantum capacitance of graphene as a function of gate potential using a three-electrode electrochemical configuration. The quantum capacitance has a non-zero minimum at the Dirac point and a linear increase on both sides of the minimum with relatively small slopes. Our findings -- which are not predicted by theory for ideal graphene -- suggest that charged impurities also influences the quantum capacitance. We also measured the capacitance in aqueous solutions at different ionic concentrations, and our results strongly indicate that the long-standing puzzle about the interfacial capacitance in carbon-based electrodes has a quantum origin.