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As a PhD thesis project, an effort is underway to observationally determine the primordial abundance of deuterium, a quantity which is poorly tested at present. This abundance is a key prediction of the hot big bang cosmology, but only galactic detections of deuterium have been able to test the theory so far. Since D is believed to be destroyed during stellar evolution, galactic data only yield a lower limit to the primordial D abundance. This project attempts to measure the cosmic abundance of deuterium at high redshifts, by looking for D absorption in bright QSO spectra. Since it is expected that the D/H ratio is no larger than $10^{-4}$, we need H column densities of at least $10^{17}$ in order for the D line to show up. Absorbers of such high column density can be identified with ``Lyman-limit absorbers'', which absorb all QSO radiation beyond the Lyman-limit in the rest frame of the absorber. Observations are done with the KPNO 4m telescope, using the echelle spectrograph in a configuration which gives us 9 km/s resolution over a wavelength range of 1700 \AA. We display promising data obtained during a run in November 1992 of the QSO GC0636+680. In this case the main component of the absorber, in which most of the HI is located, lies on the blue side of the absorber complex, has a high enough column density and is sufficiently narrow that it is the best candidate site for a D search found so far. We present a spectrum which displays Ly$\alpha$, $\beta$, and $\gamma$ absorption and a tentative indication of D Ly$\alpha$. A reliable determination of the D abundance will only be possible with better signal-to-noise data, which can be obtained with a 4m class telescope.
