We present experimental evidence of charge exchange between laser-cooled potassium ³⁹K atoms and calcium ⁴⁰Ca⁺ ions in a hybrid atom-ion trap and give quantitative theoretical explanations for the observations. The ³⁹K atoms and ⁴⁰Ca⁺ ions are held in a magneto-optical (MOT) and a linear Paul trap, respectively. Fluorescence detection and high resolution time of flight mass spectra for both species are used to determine the remaining number of ⁴⁰Ca⁺ ions, the increasing number of ³⁹K⁺ ions, and ³⁹K number density as functions of time. Simultaneous trap operation is guaranteed by alternating periods of MOT and ⁴⁰Ca⁺ cooling lights, thus avoiding direct ionization of ³⁹K by the ⁴⁰Ca⁺ cooling light. We show that the K-Ca⁺ charge-exchange rate coefficient increases linearly from zero with ³⁹K number density and the fraction of ⁴⁰Ca⁺ ions in the 4p ²P_1/2 electronically-excited state. Combined with our theoretical analysis, we conclude that these data can only be explained by a process that starts with a potassium atom in its electronic ground state and a calcium ion in its excited 4p ²P_1/2 state producing ground-state ³⁹K⁺ ions and metastable, neutral Ca (3d4p ³P₁) atoms, releasing only 150 cm^-1 equivalent relative kinetic energy. Charge-exchange between either ground- or excited-state ³⁹K and ground-state ⁴⁰Ca⁺ is negligibly small as no energetically-favorable product states are available. Our experimental and theoretical rate coefficients are in agreement given the uncertainty budgets.