We present the design of a 100 kW, J-band, third harmonic large-orbit gyrotron amplifier which utilizes the interaction between a 45 kV, 4 A beam and a vane resonator output cavity operating in the "?r"-mode. An efficiency of 55% is predicted with a large signal gain near 20 dB by a single particle code which takes into account nonideal effects associated with finite beam thickness and finite magnetic field transition widths. High efficiency is achieved by velocity modulation of an axially-streaming annular beam via a short TM310 drive cavity. Ballistically created axial bunches are converted into azimuthal bunches when the beam encounters a nonadiabatic, balanced magnetic field reversal at the end of a 30 cm drift region. The design of this tube is presented and its performance is completely characterized before the prospects for the operation of this low voltage configuration at other harmonics are explored.