  We report NuSTAR observations of a sample of six X-ray weak broad absorption
line (BAL) quasars. These targets, at z=0.148-1.223, are among the optically
brightest and most luminous BAL quasars known at z<1.3. However, their
rest-frame 2 keV luminosities are 14 to >330 times weaker than expected for
typical quasars. Our results from a pilot NuSTAR study of two low-redshift BAL
quasars, a Chandra stacking analysis of a sample of high-redshift BAL quasars,
and a NuSTAR spectral analysis of the local BAL quasar Mrk 231 have already
suggested the existence of intrinsically X-ray weak BAL quasars, i.e., quasars
not emitting X-rays at the level expected from their optical/UV emission. The
aim of the current program is to extend the search for such extraordinary
objects. Three of the six new targets are weakly detected by NuSTAR with <45
counts in the 3-24 keV band, and the other three are not detected. The hard
X-ray (8-24 keV) weakness observed by NuSTAR requires Compton-thick absorption
if these objects have nominal underlying X-ray emission. However, a soft
stacked effective photon index ({\Gamma}~1.8) for this sample disfavors
Compton-thick absorption in general. The uniform hard X-ray weakness observed
by NuSTAR for this and the pilot samples selected with <10 keV weakness also
suggests that the X-ray weakness is intrinsic in at least some of the targets.
We conclude that the NuSTAR observations have likely discovered a significant
population (>33%) of intrinsically X-ray weak objects among the BAL quasars
with significantly weak <10 keV emission. We suggest that intrinsically X-ray
weak quasars might be preferentially observed as BAL quasars.
