  We present analysis and spectral fits of the RXTE data from the May/June 1999
outburst of Aql X-1. These data include observations in the rising portion of
the hard state, in the soft state, and in the falling portion of the hard
state. We show that the data can be fit by a purely thermal Comptonization
model for all the observations, but that more complicated models cannot be
ruled out. Up to 60% of the corona's power in the soft state may be injected
into non-thermal electrons. The soft state data show approximately constant
optical depth and coronal temperatures over a range of $\sim10$ in luminosity,
while they show evidence for a reduction of seed photon temperature with
reduced luminosity and indicate that the characteristic size of the seed photon
emitting region is roughly constant throughout the soft state. The hard state
before the soft state shows a higher luminosity, higher optical depth, and
lower electron temperature than the hard state after the soft state. We find a
reduction of the hard (30-60 keV) X-ray flux during a type I burst and show
that it requires a total corona energy reservoir of less than $\sim 10^{38}$
ergs.
