We propose a new triple-junction solar cell structure composed of a III-V heterojunction bipolar transistor solar cell (HBTSC) stacked on top of, and series-connected to, a Si solar cell (III-V-HBTSC-on-Si). The HBTSC is a novel three-terminal device, whose viability has been recently experimentally demonstrated. It has the theoretical efficiency limit of an independently-connected double-junction solar cell. Here, we perform detailed balance efficiency limit calculations under one-sun illumination that show that the absolute efficiency limit of a III-V-HBTSC-on-Si device is the same as for the conventional current-matched III-V-on-Si triple-junction (47% assuming black-body spectrum, 49% with AM1.5G). However, the range of band-gap energies for which the efficiency limit is above 40% is much wider in the III-V-HBTSC-on-Si stack case. From a technological point of view, the lattice-matched GaInP/GaAs combination is particularly interesting, which has an AM1.5G efficiency limit of 47% with the HBTSC-on-Si structure and 39% if the current-matched III-V-on-Si triple junction is considered. Moreover, we show that interconnecting the terminals of the HBTSC to achieve a two-terminal GaInP/GaAs-HBTSC-on-Si device only reduces the efficiency limit by three points, to 43%. As a result, the GaInP/GaAs-HBTSC-on-Si solar cell becomes a promising device for two-terminal, high-efficiency one-sun operation. For it to also be cost-effective, low-cost technologies must be applied to the III-V material growth, such as high-throughput epitaxy or sequential growth.