  The total cross section of the $^{82}$Kr(p,$\gamma$)$^{83}$Rb reaction was
measured for the first time at effective center-of-mass energies between 2.4
and 3.0 MeV, within the relevant Gamow window for the astrophysical $\gamma$
process. The experiment took place at the National Superconducting Cyclotron
Laboratory at Michigan State University using the ReA facility. A $^{82}$Kr
beam was directed onto a hydrogen gas cell located at the center of the Summing
NaI(Tl) (SuN) detector. The obtained spectra were analyzed using the
$\gamma$-summing technique and the extracted cross section was compared to
standard statistical model calculations using the \textsc{non-smoker} and
\textsc{talys} codes. The comparison indicates that standard statistical model
calculations tend to overproduce the cross section of the
$^{82}$Kr(p,$\gamma$)$^{83}$Rb reaction relative to the experimentally measured
values. Furthermore, the experimental data was used to provide additional
constraints on the nuclear level density and $\gamma$-ray strength function
used in the statistical model calculations.
