Accurate absorption spectra of gases in the near-ultraviolet (300 to 400 nm) are essential in atmospheric observations and laboratory studies. This paper describes a novel incoherent broadband cavity-enhanced absorption spectroscopy (IBBCEAS) instrument for measuring very weak absorption spectra from 335 to 375 nm. The instrument performance was validated against the B-3(1)-X(1)A(1) transition of SO2. The measured absorption varied linearly with SO2 column density and the resulting spectrum agrees well with published spectra. Using the instrument, we report new absorption cross-sections of O-3, acetone, 2-butanone, and 2-pentanone in this spectral region, where literature data diverge considerably. In the absorption minimum between the Huggins and Chappuis bands, our absorption spectra fall at the lower range of reported ozone absorption cross-sections. The spectra of the ketones agree with prior spectra at moderate absorptions, but differ significantly at the limits of other instruments' sensitivity. The collision-induced absorption of the O-4 dimer at 360.5 nm was also measured and found to have a maximum cross-section of ca. 4.0 x 10(-46) cm(5) molecule(-2). We demonstrate the application of the instrument to quantifying low concentrations of the short-lived radical, BrO, in the presence of stronger absorptions from Br-2 and O-3.