We report a continuous-wave, doubly resonant optical parametric oscillator (OPO) based on the nonlinear material periodically poled KTiOPO4 and its application to spectroscopy. The OPO, which is pumped by a diode-pumped frequency-doubled Nd:YLF laser at 523 nm, has a low pump-power threshold of 25 mW and can deliver 10 mW of single-frequency output at 1.65 mu m for a pump power of 200 mW. The idler wavelength can be temperature tuned at a rate of 0.73 nm/degrees C, and smooth tuning of the output frequency over similar to 3 GHz is achieved by smooth tuning of the pump laser. We demonstrate the practicality of the OPO by recording the absorption spectrum of methane near 1649 nm.