We use a rapidly swept optical cavity in a novel approach to cavity ringdown spectroscopy (CRDS) with a continuous-wave (cw) laser. This simplifies cw-CRDS by eliminating the need for a fast optical switch and is demonstrated by recording weak combination-band spectra of carbon dioxide diluted in nitrogen, using a cw single-mode tunable diode laser (TDL) in the 1.55 mu m near-infrared region. We employ a simultaneous combination of CRDS and cavity-enhanced absorption spectroscopy, to compensate for nonlinear CRDS response that is observed for stronger absorption features. Experimental results and performance characteristics are presented, together with numerical simulations.