High-resolution and high-sensitivity molecular spectroscopy is widely used in fundamental molecular physics, atmospheric studies, remote sensing, industrial process monitoring, and medical diagnostics. Accurate determination of the parameters of molecule absorption lines, such as line positions, line strengths, line widths and profiles, is essential to support these studies and applications. For example, in order to retrieve the column density of carbon dioxide with a precision of one part per million (ppm), we need laboratory data of line positions with a uncertainty lower than 0.3 MHz and line intensities with a relative accuracy better than 0.5%. Here we present precision spectroscopy of molecules using a laser locked with a high-finesse cavity. The cavity made of invar is thermo-stabilized to reduce the drifts of its length and the cavity mode frequencies. The frequency of the probe laser is locked on a longitudinal mode of the cavity by using the Pound-Drever-Hall method. Another beam from the probe laser, which is frequency shifted and on resonance with a nearby longitudinal mode of the cavity, is used for cavity ring-down spectrum (CRDS) measurement. The CRDS absorption spectrum is recorded by stepping the modulation frequency of a fiber electro-optic modulator in increment of the mode spacing of the cavity. Note that the cavity mode frequencies are shifted due to the dispersion introduced by the absorption lines. Prior to the CRDS measurements, the transmittance spectra of the cavity modes are recorded by scanning the probe laser frequencies over the resonance, which allows the determination of the cavity mode frequencies with an accuracy at a Hz level. Therefore, a dispersion spectrum is also obtained using the same setup by measuring the frequency shifts of cavity modes of the samples with and without absorption. The absolute frequency of the probe laser is determined by an optical frequency comb referring to a GPS-disciplined rubidium clock. The long term drift of beat frequency between the optical frequency comb and the probe laser is measured to be about 1.8 MHz per hour, which is consistent with the thermal expansion of the cavity under a temperature drift of 50 mK. The performance of the spectrometer is demonstrated by measuring the Doppler-broadened spectra of CO2 around 6470.42 cm-1. Precise spectroscopic parameters are derived from both the absorption and dispersion spectra recorded by the same spectrometer. The line position is determined with an accuracy of 0.18 MHz, which is over one order of magnitude better than those given in previous studies and spectral databases.