A possible manifestation of an additional light gauge boson A', named a dark photon, associated with a group U(1)(B-L), is studied in neutrino-electron scattering experiments. The exclusion plot on the coupling constant g(B-L) and the dark photon mass M-A' is obtained. It is shown that the contributions of interference terms between the dark photon and the Standard Model are important. The interference effects are studied and compared with data sets from TEXONO, GEMMA, BOREXINO, and LSND, as well as CHARM II experiments. Our results provide more stringent bounds to some regions of parameter space.