Recent results from the ROSAT All-Sky Survey ( RASS ) 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 BALQSOs observed within the inner 20 ^ { \prime } of the ROSAT PSPC . This sample includes both targeted and serendipitous exposures ranging from 8 to 75 ksec . Despite these deep exposures , most of the BALQSOs are undetected , and have unusually weak X-ray emission , as evidenced by large optical-to-X-ray slopes \alpha _ { ox } . Large values of \alpha _ { ox } ( > \atop { \sim } 1.8 ) may prove to be a defining characteristic of BALQSOs . We predict that samples of QSO candidates with large \alpha _ { ox } will yield a higher percentage of BALQSOs , particularly at low redshift . As a corollary , X-ray-selected QSO samples should yield fewer BALQSOs . 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 BALQSOs . To constrain the total absorbing column of the BAL clouds , we compare our measured soft X-ray fluxes or upper limits to those expected from normal radio quiet QSOs of comparable optical continuum magnitude and redshift . From sensitive X-ray observations , we derive column densities of > \atop { \sim } 2 \times 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 .