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Many quantum-information applications require high-efficiency, low-noise, single-photon detectors that operate at visible and near-infrared wavelengths. The tunable superconducting critical temperature and anomalously low electron-phonon coupling of tungsten make it a suitable material for the fabrication of transition-edge sensors (TESs) that meet these requirements. The quantum efficiency of a typical tungsten TES detector, intrinsically around 15% at 1550 nm, can be increased by placing the tungsten detector in a resonant cavity, but the performance of a device embedded in a cavity has not been tested previously. We demonstrate that the presence of the cavity does not adversely affect the sensitivity or response time, and we report on the device characteristics of a new generation of tungsten TESs with greater than 80% quantum efficiency at 1550 nm.