Recent ROSAT studies have identified a significant population of Active Galactic Nuclei (AGN) that are notably faint in soft X-rays relative to their optical fluxes. Are these AGN intrinsically X-ray weak or are they just highly absorbed? Brandt, Laor & Wills have systematically examined the optical and UV spectral properties of a well-defined sample of these soft X-ray weak (SXW) AGN drawn from the Boroson & Green sample of all the Palomar Green AGN with z<0.5. We present ASCA observations of three of these SXW AGN: PG 1011-040, PG 1535+547 (Mrk 486), and PG 2112+059. In general, our ASCA observations support the intrinsic absorption scenario for explaining soft X-ray weakness; both PG 1535+547 and PG 2112+059 show significant column densities (N_H~10^22-10^23 cm^-2) of absorbing gas. Interestingly, PG 1011-040 shows no spectral evidence for X-ray absorption. The weak X-ray emission may result from very strong absorption of a partially covered source, or this AGN may be intrinsically X-ray weak. PG 2112+059 is a Broad Absorption Line (BAL) QSO, and we find it to have the highest X-ray flux known of this class. It shows a typical power-law X-ray continuum above 3 keV; this is the first direct evidence that BAL QSOs indeed have normal X-ray continua underlying their intrinsic absorption. Finally, marked variability between the ROSAT and ASCA observations of PG 1535+547 and PG 2112+059 suggests that the soft X-ray weak designation may be transient, and multi-epoch 0.1-10.0 keV X-ray observations are required to constrain variability of the absorber and continuum.