  We present a cosmic microwave background (CMB) lensing map produced from a
linear combination of South Pole Telescope (SPT) and \emph{Planck} temperature
data. The 150 GHz temperature data from the $2500\ {\rm deg}^{2}$ SPT-SZ survey
is combined with the \emph{Planck} 143 GHz data in harmonic space, to obtain a
temperature map that has a broader $\ell$ coverage and less noise than either
individual map. Using a quadratic estimator technique on this combined
temperature map, we produce a map of the gravitational lensing potential
projected along the line of sight. We measure the auto-spectrum of the lensing
potential $C_{L}^{\phi\phi}$, and compare it to the theoretical prediction for
a $\Lambda$CDM cosmology consistent with the \emph{Planck} 2015 data set,
finding a best-fit amplitude of $0.95_{-0.06}^{+0.06}({\rm Stat.})\!
_{-0.01}^{+0.01}({\rm Sys.})$. The null hypothesis of no lensing is rejected at
a significance of $24\,\sigma$. One important use of such a lensing potential
map is in cross-correlations with other dark matter tracers. We demonstrate
this cross-correlation in practice by calculating the cross-spectrum,
$C_{L}^{\phi G}$, between the SPT+\emph{Planck} lensing map and Wide-field
Infrared Survey Explorer (\emph{WISE}) galaxies. We fit $C_{L}^{\phi G}$ to a
power law of the form $p_{L}=a(L/L_{0})^{-b}$ with $a=2.15 \times 10^{-8}$,
$b=1.35$, $L_{0}=490$, and find $\eta^{\phi G}=0.94^{+0.04}_{-0.04}$, which is
marginally lower, but in good agreement with $\eta^{\phi
G}=1.00^{+0.02}_{-0.01}$, the best-fit amplitude for the cross-correlation of
\emph{Planck}-2015 CMB lensing and \emph{WISE} galaxies over $\sim67\%$ of the
sky. The lensing potential map presented here will be used for
cross-correlation studies with the Dark Energy Survey (DES), whose footprint
nearly completely covers the SPT $2500\ {\rm deg}^2$ field.
