Two coherent frequency combs are used to measure the full complex response of a sample in a configuration analogous to a dispersive Fourier transform spectrometer, infrared time domain spectrometer, or a multiheterodyne laser spectrometer. This dual comb spectrometer retains the frequency accuracy and resolution of the reference underlying the stabilized combs. We discuss the specific design of our coherent dual-comb spectrometer and demonstrate the potential of this technique by measuring the overtone vibration of hydrogen cyanide, centered at 194 THz (1545 nm). We measure the fully normalized, complex response of the gas over a 9 THz bandwidth at 220 MHz frequency resolution yielding 41,000 resolution elements. The average spectral signal-to-noise ratio (SNR) over the 9 THz bandwidth is 2,500 for both the magnitude and phase of the measured spectral response and the peak SNR is 4,000. This peak SNR corresponds to a fractional absorption sensitivity of 0.05% and a phase sensitivity of 250 microradians. As the spectral coverage of combs expands, coherent dual-comb spectroscopy could provide high frequency accuracy and resolution measurements of a complex sample response across a range of spectral regions.Work of U.S. government, not subject to copyright.