  The physical conditions in the inner parsec of accretion disks believed to
orbit the central black holes in active galactic nuclei can be probed by
imaging the absorption (by ionized gas in the disk) of background emission from
a radio counterjet. We report high angular resolution VLBI observations of the
nearby (about 40 Mpc) radio galaxy NGC 4261 that confirm free-free absorption
of radio emission from a counterjet by a geometrically thin, nearly edge-on
disk at 1.6, 4.8, and 8.4 GHz. The angular width and depth of the absorption
appears to increase with decreasing frequency, as expected. We derive an
average electron density of ~10E4 per cc at a disk radius of about 0.2 pc,
assuming that the inner disk inclination and opening angles are the same as at
larger radii. Pressure balance between the thermal gas and the magnetic field
in the disk implies an average field strength of 0.1 milligauss at a radius of
0.2 pc. These are the closest-in free-free absorption measurements to date of
the conditions in an extragalactic accretion disk orbiting a black hole with a
well-determined mass. If a standard advection-dominated accretion flow exists
in the disk center, then the transition between thin and thick disk regions
must occur at a radius less than 0.2 pc (4000 Schwarzschild radii).
