Cavity-enhanced absorption spectroscopy is promising for many applications requiring a very high concentration sensitivity but often accompanied by low temporal resolution. In this Letter, we demonstrate a broadband cavity-enhanced absorption spectrometer capable of detection rates of up to 50 kHz, based on a spatially coherent supercontinuum (SC) light source and an in-house-built, high-speed near-infrared spectrograph. The SC spectrometer allows for the simultaneous quantitative detection of CO2, C2H2, and H2O within a spectral range from 1420 to 1570 nm. Using cavity mirrors with a specified reflectivity of R=98.0±0.3% a minimal spectrally averaged absorption coefficient of αmin=1·10-5 cm-1 can be detected at a repetition rate of 50 kHz.