We present combined \approx 14 \textrm { - - } 37 ~ { } ks Chandra observations of seven z = 1.6 \textrm { - - } 2.7 broad absorption line ( BAL ) quasars selected from the Large Bright Quasar Survey ( LBQS ) . These seven objects are high-ionization BAL ( HiBAL ) quasars , and they were undetected in the Chandra hard band ( 2–8 keV ) in previous observations . The stacking analyses of previous Chandra observations suggested that these seven objects likely contain some candidates for intrinsically X-ray weak BAL quasars . With the new Chandra observations , six targets are detected . We calculate their effective power-law photon indices and hard-band flux weakness , and find that two objects , LBQS 1203 + 1530 and LBQS 1442 - 0011 , show soft/steep spectral shapes ( \Gamma _ { eff } = 2.2 ^ { +0.9 } _ { -0.9 } and 1.9 _ { -0.8 } ^ { +0.9 } ) and significant X-ray weakness in the hard band ( by factors of \approx 15 and 12 ) . We conclude that the two HiBAL quasars are good candidates for intrinsically X-ray weak BAL quasars . The mid-infrared-to-UV spectral energy distributions ( SEDs ) of the two candidates are consistent with those of typical quasars . We constrain the fraction of intrinsically X-ray weak AGNs among HiBAL quasars to be \approx 7 \textrm { - - } 10 \% ( 2/29–3/29 ) , and we estimate it is \approx 6 \textrm { - - } 23 \% ( 2/35–8/35 ) among the general BAL quasar population . Such a fraction is considerably larger than the fraction among non-BAL quasars , and we suggest that intrinsically X-ray weak quasars are preferentially observed as BAL quasars . Intrinsically X-ray weak AGNs likely comprise a small minority of the luminous type 1 AGN population , and they should not affect significantly the completeness of these AGNs found in deep X-ray surveys .