The spectral range between 1650 and 1700 nm is an interesting region due to its potential applications in optical telecommunication and optical-based methane sensing. Unfortunately, the availability of compact and simple optical amplifiers with output powers exceeding tens of milliwatts in this spectral region is still limited. In this paper, a single-frequency continuous-wave bismuth-doped fiber amplifier (BDFA) operating at 1651 and 1687 nm is presented. With the improved signal/pump coupling and modified pump source design, the output powers of 163 mW (at 1651 nm) and 197 mW (at 1687 nm) were obtained. Application of the BDFA to the optical spectroscopy of methane near 1651 nm is also described. We demonstrate that the BDFA can be effectively used for signal amplitude enhancement in photothermal interferometry.