An oxychloride superconductor Ca2-xCuO2Cl2 with a single CuO2 plane in the unit cell was prepared without cation substitution using high-pressure synthesis. The highest T-C was 38 K, 10 K higher than that of Ca2-xNaxCuO2Cl2. Structure analysis based on synchrotron x-ray and neutron powder diffractions revealed that the Ca deficiency was the origin of the hole carrier. This compound has two structural features as compared to Ca2-xNaxCuO2Cl2; fewer defects and a shorter Cu-Cl bond length. Postannealing at 773 K led to a further increase of the T-C to 43 K. This superconductor with a flat CuO2 plane might form the basis for future discussions about the factors that determine the T-C of single-layer cuprates.