  We analyze {\it simultaneously} six composite {\it RXTE} spectra of GX 339--4
in the hard state comprising 77 million counts collected over 196 ks. The
source spectra are ordered by luminosity and spanthe range 1.6\% to 17\% of the
Eddington luminosity. Crucially, using our new tool {\tt pcacorr}, we
re-calibrate the data to a precision of 0.1\%, an order of magnitude
improvement over all earlier work. Using our advanced reflection model {\tt
relxill}, we target the strong features in the component of emission reflected
from the disk, namely, the relativistically-broadened Fe K emission line, the
Fe K edge and the Compton hump. We report results for two joint fits to the six
spectra: For the first fit, we fix the spin parameter to its maximal value
($a_*=0.998$) and allow the inner disk radius $R_{\rm in}$ to vary. Results
include (i) precise measurements of $R_{\rm in}$, with evidence that the disk
becomes slightly truncated at a few percent of Eddington; and (ii) an
order-of-magnitude swing with luminosity in the high energy cutoff, which
reaches $>890$ keV at our lowest luminosity. For the second fit, we make the
standard assumption in estimating spin that the inner edge of the accretion
disk is located at the innermost stable circular orbit ($R_\mathrm{in} =
R_\mathrm{ISCO}$) and find $a_* = 0.95^{+0.03}_{-0.05}$ (90\% confidence,
statistical). For both fits, and at the same level of statistical confidence,
we estimate that the disk inclination is $i = 48\pm 1$ deg and that the Fe
abundance is super-solar, $A_\mathrm{Fe} = 5\pm1$.
