  We present the second multi-frequency radio detection of a reverse shock in a
$\gamma$-ray burst. By combining our extensive radio observations of the
Fermi-LAT GRB 160509A at $z = 1.17$ up to $20$ days after the burst with Swift
X-ray observations and ground-based optical and near-infrared data, we show
that the afterglow emission comprises distinct reverse shock and forward shock
contributions: the reverse shock emission dominates in the radio band at
$\lesssim10~$days, while the forward shock emission dominates in the X-ray,
optical, and near-infrared bands. Through multi-wavelength modeling, we
determine a circumburst density of $n_0\approx10^{-3}~$cm$^{-3}$, supporting
our previous suggestion that a low-density circumburst environment is conducive
to the production of long-lasting reverse shock radiation in the radio band. We
infer the presence of a large excess X-ray absorption column, $N_{\rm H}
\approx 1.5\times10^{22}~$cm$^{-2}$, and a high rest-frame optical extinction,
$A_{\rm V}\approx3.4~$mag. We identify a jet break in the X-ray light curve at
$t_{\rm jet}\approx6~$d, and thus derive a jet opening angle of $\theta_{\rm
jet}\approx4~$deg, yielding a beaming-corrected kinetic energy and radiated
$\gamma$-ray energy of $E_{\rm K}\approx4\times10^{50}~$erg and
$E_{\gamma}\approx1.3\times10^{51}~$erg ($1$-$10^4~$keV, rest frame),
respectively. Consistency arguments connecting the forward and reverse shocks
suggest a deceleration time of $t_{\rm dec} \approx 460~$s$~\approx T_{90}$, a
Lorentz factor of $\Gamma(t_{\rm dec})\approx330$, and a reverse shock to
forward shock fractional magnetic energy density ratio of $R_{\rm
B}\equiv\epsilon_{\rm B,RS}/\epsilon_{\rm B,FS}\approx8$.
