A theory of the electrical double layer in high-temperature superconductors is developed in the context of a modified Ginzburg-Landau equation. Potential and excess charge distributions inside the electrode are computed. These distributions are shown to be characterized by a new fundamental length parameter. This length diverges when the electrode temperature approaches the superconducting critical temperature. The capacitance of the electrical double layer is found to be inversely proportional to this length. On the basis of this analysis, the origin of an experimentally observed dip in the double-layer capacitance is explained. The dependence of this capacitance dip on the external field is predicted.