We demonstrate quantum-noise-limited frequency-comb cavity-enhanced absorption spectroscopy with spectrally multiplexed detection for acquisition times ranging from 10 ms up to 10 min, where a record absorption sensitivity of 7×10-13 cm-1 per spectral element is attained with nW power levels. For this, a widely tuneable frequency-doubled free-running fs Ti:sapphire laser oscillator is coupled to a cavity of extreme finesse (32 000) whose length is modulated and whose output is spectrally dispersed over a linear charge coupled device array. This scheme is robust, as it does not require tight frequency locking or controlling the comb frequency offset, and works with spectrally broad combs, as it is not limited by cavity dispersion.