Thermophotovoltaic cells are similar to solar cells, but instead of converting solar radiation to electricity, they are designed to utilize locally radiated heat. Development of high-efficiency thermophotovoltaic cells has the potential to enable widespread applications in grid-scale thermal energy storage(1,2), direct solar energy conversion(3-8), distributed co-generation(9-11)and waste heat scavenging(12). To reach high efficiencies, thermophotovoltaic cells must utilize the broad spectrum of a radiative thermal source. However, most thermal radiation is in a low-energy wavelength range that cannot be used to excite electronic transitions and generate electricity. One promising way to overcome this challenge is to have low-energy photons reflected and re-absorbed by the thermal emitter, where their energy can have another chance at contributing towards photogeneration in the cell. However, current methods for photon recuperation are limited by insufficient bandwidth or parasitic absorption, resulting in large efficiency losses relative to theoretical limits. Here we demonstrate near-perfect reflection of low-energy photons by embedding a layer of air (an air bridge) within a thin-film In0.53Ga0.47As cell. This result represents a fourfold reduction in parasitic absorption relative to existing thermophotovoltaic cells. The resulting gain in absolute efficiency exceeds 6 per cent, leading to a very high power conversion efficiency of more than 30 per cent, as measured with an approximately 1,455-kelvin silicon carbide emitter. As the out-of-band reflectance approaches unity, the thermophotovoltaic efficiency becomes nearly insensitive to increasing cell bandgap or decreasing emitter temperature. Accessing this regime may unlock a range of possible materials and heat sources that were previously inaccessible to thermophotovoltaic energy conversion. An air gap embedded within the structure of a thermophotovoltaic device acts as a near-perfect reflector of low-energy photons, resulting in their recovery and recycling by the thermal source, enabling excellent power-conversion efficiency.