A compact noise-immune cavity-enhanced optical heterodyne molecular spectrometry (NICE-OHMS) spectrometer, based on a narrowband erbium-doped fiber laser and an integrated optics electro-optic modulator, has been developed for trace species detection. A general theoretical description of NICE-OHMS signals demodulated at an arbitrary FM detection phase is provided in terms of the analyte concentration. Explicit expressions for Doppler-broadened NICE-OHMS line shapes, which are in excellent agreement with the measurements, are given. In a first demonstration, using a cavity with a finesse of 1400, acetylene has been detected on a Doppler-broadened transition at $\sim 1531\mathrm{nm}$ . A limit of detection of $130\mathrm{ppt}$ , corresponding to an absorption of $2.4\times {10}^{-9}{\mathrm{cm}}^{-1}$ , has been obtained.