A High Sensitivity Absorption experiment which is used to measure UV and VUV oscillator strengths of both atoms and ions has been developed at the University of Wisconsin-Madison. This experiment incorporates a hollow cathode discharge as an absorbing sample, the Aladdin storage ring at the Synchrotron Radiation Center as a continuum source, and a 3 m focal length vacuum echelle spectrometer equipped with a CCD detector array. The experiment achieves spectral resolving powers of 350,000 and sensitivities to fractional absorptions much smaller than 1% at deep UV and VUV wavelengths. Although absorption techniques have long been used in measuring oscillator strengths, detector arrays vastly increase the sensitivity, and synchrotron radiation extends the wavelength coverage, of such experiments. Column densities of sputtered metal atoms and ions as small as w3.0x10(8) cm(-2) can be detected. Our experiment is used to measure relative oscillator strengths for lines from a common lower level, usually a VUV transition relative to a well known UV transition. An accurate absolute scale for all of these measurements is established using radiative lifetimes from laser induced fluorescence measurements on atomic/ionic beams. This experiment is applicable to essentially every element in the periodic table, both neutral atoms and atomic ions. The experiment is used to measure oscillator strengths for the most important UV and VUV resonance lines of Fe+, Co+, and Ni+. These resonance lines are prominent in absorption spectra on the Interstellar Medium (ISM) recorded using the Hubble Space Telescope.