We experimentally investigate the discrete mode in the zero-average-refractive-index gap (the zero-n—gap) of the one-dimensional periodical structures containing left-handed materials (LHMs) and right-handed materials (RHMs). When the discrete mode conditions are satisfied in the zero-n gap, there are no discrete propagating modes, but the zero-n—gap completely vanishes. This behavior is attributed to the dispersive property of LHMs and RHMs. In addition, the theoretical expectation of the discrete modes with no sidelobes in the zero-n—gap never occurs in experiment because the LHMs is intrinsically dispersive.