We have performed the first direct measurement of the ^{83}Rb(p,γ) radiative capture reaction cross section in inverse kinematics using a radioactive beam of ^{83}Rb at incident energies of 2.4 and 2.7A MeV. The measured cross section at an effective relative kinetic energy of E_{cm}=2.393 MeV, which lies within the relevant energy window for core collapse supernovae, is smaller than the prediction of statistical model calculations. This leads to the abundance of ^{84}Sr produced in the astrophysical p process being higher than previously calculated. Moreover, the discrepancy of the present data with theoretical predictions indicates that further experimental investigation of p-process reactions involving unstable projectiles is clearly warranted.