  We present new spectral line observations of the CH3CN molecule in the
accretion disk around the massive protostar IRAS 20126+4104 with the
Submillimeter Array that for the first time measure the disk density,
temperature, and rotational velocity with sufficient resolution (0.37",
equivalent to ~600 AU) to assess the gravitational stability of the disk
through the Toomre-Q parameter. Our observations resolve the central 2000 AU
region that shows steeper velocity gradients with increasing upper state
energy, indicating an increase in the rotational velocity of the hotter gas
nearer the star. Such spin-up motions are characteristics of an accretion flow
in a rotationally supported disk. We compare the observed data with synthetic
image cubes produced by three-dimensional radiative transfer models describing
a thin flared disk in Keplerian motion enveloped within the centrifugal radius
of an angular-momentum-conserving accretion flow. Given a luminosity of
1.3x10^4 Lsun, the optimized model gives a disk mass of 1.5 Msun and a radius
of 858 AU rotating about a 12.0 Msun protostar with a disk mass accretion rate
of 3.9x10^{-5} Msun/yr. Our study finds that, in contrast to some theoretical
expectations, the disk is hot and stable to fragmentation with Q > 2.8 at all
radii which permits a smooth accretion flow. These results put forward the
first constraints on gravitational instabilities in massive protostellar disks,
which are closely connected to the formation of companion stars and planetary
systems by fragmentation.
