We have demonstrated an electrically conductive ZnO/GaN multilayer structure using hybrid plasma-assisted molecular beam epitaxy. Electrical I-V characteristics were measured through the top three pairs of a six pair ZnO/GaN sample. The total measured resistance was dominated by lateral and contact resistances, setting an upper limit of ~10 ^-4 Ω·cm ² for the vertical specific series resistance of the stack. A strong contribution to the low resistance is the cancellation of spontaneous and piezoelectric polarization that occurs in the in-plane strained ZnO/GaN sample, as shown by electrical simulations. In addition, the simulations show that the actual vertical resistance of the sample could in fact be three orders of magnitude lower and that ZnO/GaN structures with thicknesses fulfilling the Bragg condition should have similar resistance. Our results suggest that ZnO/GaN distributed Bragg reflectors (DBRs) are a promising alternative to pure III-nitride DBRs in GaN-based vertical-cavity surface-emitting lasers.