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Recent results from the ROSAT All-Sky Survey have shown that broad absorption line (BAL) QSOs are either highly absorbed or underluminous in the soft X-ray bandpass. Here we extend this work by analyzing all known bona fide BAL QSOs observed within the inner 20' of the ROSAT Position Sensitive Proportional Counter. This sample includes both targeted and serendipitous exposures ranging from 8 to 75 ks. Despite these deep exposures, most of the BAL QSOs are undetected and have unusually weak X- ray emission, as evidenced by large optical-to-X-ray slopes α_OX_. Large values of α_OX_ (~>1.8) may prove to be a defining characteristic of BAL QSOs. We predict that samples of QSO candidates with large α_OX_ will yield a higher percentage of BAL QSOs, particularly at low redshift. As a corollary, X-ray-selected QSO samples should yield fewer BAL QSOs. The optical/UV emission line spectra of BAL and non-BAL QSOs are quite similar, suggesting that their intrinsic spectral energy distributions are similar as well. Absorption thus seems the likely reason for the X- ray-quiet nature of BAL QSOs. To constrain the total absorbing column of the BAL clouds, we compare our measured soft X-ray fluxes or upper limits with those expected from normal radio-quiet QSOs of comparable optical continuum magnitude and redshift. From sensitive X-ray observations, we derive column densities of ~> 2 x 10^22^ cm^-2^ for intrinsic cold absorbers of solar metallicity. These new results suggest columns at least an order of magnitude larger than the columns previously estimated from optical/UV spectra alone.