A novel approach to cavity ringdown spectroscopy uses a continuous-wave laser and a rapidly swept optical cavity to shift the frequency of optical radiation stored in the cavity. This frequency-shifted radiation from the ringdown cavity is then combined with incident laser radiation to generate optical heterodyne signals, simply and efficiently. A noise-limited absorbance sensitivity of 3 x 10(-9) cm(-1) is realised, using similar to 35 muW of single-mode radiation from a 1.53 mum tunable diode laser. The resonance properties of a swept optical cavity simplify this heterodyne-detected technique by avoiding the customary need for a fast optical switch or for wavelength-locking of cavity length.