We analyze the far-ultraviolet (FUV) spectra of 33 classical T Tauri stars (CTTS), including 20 new spectra obtained with the Advanced Camera for Surveys Solar Blind Channel (ACS/SBC) on the Hubble Space Telescope. Of the sources, 28 are in the ~1 Myr old Taurus-Auriga complex or Orion Molecular Cloud, 4 in the 8-10 Myr old Orion OB1a complex, and 1, TW Hya, in the 10 Myr old TW Hydrae Association. We also obtained FUV ACS/SBC spectra of 10 non-accreting sources surrounded by debris disks with ages between 10 and 125 Myr. We use a feature in the FUV spectra due mostly to electron impact excitation of H_2 to study the evolution of the gas in the inner disk. We find that the H_2 feature is absent in non-accreting sources, but is detected in the spectra of CTTS and correlates with accretion luminosity. Since all young stars have active chromospheres which produce strong X-ray and UV emission capable of exciting H_2 in the disk, the fact that the non-accreting sources show no H_2 emission implies that the H_2 gas in the inner disk has dissipated in the non-accreting sources, although dust (and possibly gas) remains at larger radii. Using the flux at 1600 Å, we estimate that the column density of H_2 left in the inner regions of the debris disks in our sample is less than ~3 × 10^(–6) g cm^(-2), 9 orders of magnitude below the surface density of the minimum mass solar nebula at 1 AU
