  We present a detailed spectral analysis of the Black Hole Binary XTE
J1752-223 in the hard state of its 2009 outburst. Regular monitoring of this
source by RXTE provided high signal-to-noise spectra along the outburst rise
and decay. During one full month this source stalled at $\sim$30\% of its peak
count rate at a constant hardness and intensity. By combining all the data in
this exceptionally-stable hard state, we obtained an aggregate PCA spectrum
(3-45 keV) with 100 million counts, and a corresponding HEXTE spectrum (20-140
keV) with 5.8 million counts. Implementing a version of our reflection code
with a physical model for Comptonization, we obtain tight constraints on
important physical parameters for this system. In particular, the inner
accretion disk is measured very close in, at $R_\mathrm{in}=1.7\pm0.4$ $R_g$.
Assuming $R_\mathrm{in}=R_\mathrm{ISCO}$, we find a relatively high black hole
spin ($a_*=0.92\pm0.06$). Imposing a lamppost geometry, we obtain a low
inclination ($i=35\pm4$ deg), which agrees with the upper limit found in the
radio ($i<49$ deg). However, we note that this model cannot be statistically
distinguished from a non-lamppost model with free emissivity index, for which
the inclination is markedly higher. Additionally, we find a relatively cool
corona ($57-70$ keV), and large iron abundance ($3.3-3.7$ solar). We further
find that properly accounting for Comptonization of the reflection emission
improves the fit significantly and causes an otherwise low reflection fraction
($\sim 0.2-0.3$) to increase by an order of magnitude, in line with geometrical
expectations for a lamppost corona. We compare these results with similar
investigations reported for GX 339-4 in its bright hard state.
