  We present the analysis and interpretation of a sample of eight ASCA
observations of Broad Absorption Line Quasi-Stellar Objects (BALQSOs). This is
the first moderate-sized sample of sensitive BALQSO observations above 2 keV,
and the BALQSOs in our sample are among the optically brightest known
(B=14.5-18.5). Despite the ability of 2-10 keV X-rays to penetrate large column
densities, we find BALQSOs to be extremely weak sources above 2 keV, and we are
only able to add two new 2-10 keV detections (0226-104 and IRAS 07598+6508) to
those previously reported. By comparison with non-BALQSOs of similar optical
continuum magnitudes, we derive the column densities needed to suppress the
expected X-ray fluxes of our BALQSOs. In several cases we derive column
densities > 5x10^{23} cm^{-2} for a neutral absorber with solar abundances.
These are the largest X-ray column densities yet inferred for BALQSOs, and they
exceed ROSAT lower limits by about an order of magnitude. Optical brightness
does not appear to be a good predictor of 2-10 keV brightness for BALQSOs, but
our data do suggest that the BALQSOs with high optical continuum polarizations
may be the X-ray brighter members of the class. For example, the highly
polarized object PHL 5200 appears to be unusually X-ray bright for a BALQSO
given its optical magnitude. We discuss the implications of our results for
future observations with AXAF and XMM. If the objects in our sample are
representative of the BALQSO population, precision X-ray spectroscopy of most
BALQSOs will unfortunately prove difficult in the near future.
