For electron cyclotron heating to be economically attractive for large scale tokamaks, such as ITER, the unit power of the power source, the gyrotron, should be as high as possible. For gyrotrons with hollow cavities, as are being developed in the United States and Europe, the highest power planned for a cw device is about 1 MW at 140 GHz. Higher powers are of interest but the ohmic heating in cavities without undue mode competition becomes too high. Work in the former Soviet Union has shown that the inclusion of an inner conductor in the cavity can reduce mode competition.  We report here work on the design of a coaxial cavity-based gyrotron. Given are both the relevant the results from a study of the feasibility of a 1 MW, 280 GHz gyrotron as well as more recent calculations specific to the 140 GHz device. We discuss briefly the issues of the cavity design, including mode competition and ohmic heating, the electron gun, and the output mode converter.