We report experimental observation of a normal incidence phononic band gap in one-dimensional periodic (SiO(2)/poly(methyl methacrylate)) multilayer film at gigahertz frequencies using Brillouin spectroscopy. The band gap to midgap ratio of 0.30 occurs for elastic wave propagation along the periodicity direction, whereas for inplane propagation the system displays an effective medium behavior. The phononic properties are well captured by numerical simulations. The porosity in the silica layers presents a structural scaffold for the introduction of secondary active media for potential coupling between phonons and other excitations, such as photons and electrons.