Simulations of the formation of large-scale structures predict that dark matter, low density highly ionized gas, and galaxies form 10 to 40 Mpc scale filaments. These structures are easily recognized in the distribution of galaxies. Here we use Ly alpha absorption lines to study the gas in 30 x 6 Mpc filament at cz ~ 3500km s super(-1), defined using a new catalog of nearby (cz < 10,000 km s super(-1)) galaxies, which is complete down to a luminosity of about 0.05 Llow * for the region of space analyzed here. Using Hubble Space Telescope spectra of 24 active galactic nuclei, we sample the gas in this filament. All of our sightlines pass outside the virial radius of any known filament galaxy. Within 500 kpc of the filament axis the detection rate is ~80%, but no detections are seen more than 2.1 Mpc from the filament axis. The width of the Ly alpha lines correlates with filament impact parameter and the four BLAs in our sample occur within 400 kpc of the filament axis, indicating increased temperature and/or turbulence. Comparing to simulations, we find that the recent Haardt & Madau extragalactic ionizing background predicts a factor of 3-5 too few ionizing photons. Using a more intense radiation field matches the hydrogen density profile within 2.1 Mpc of the filament axis, but the simulations still overpredict the detection rate between 2.1 and 5 Mpc from the axis. The baryonic mass inside filament galaxies is 1.4 x 10 super(13) M sub(middot in circle), while the mass of filament gas outside galaxy halos is found to be 5.2 x 10 super(13) M sub(middot in circle).