We have found experimentally that the shot noise of the tunneling current I through an undoped semiconductor superlattice is reduced with respect to the Poissonian noise value 2eI, and that the noise approaches 1/3 of that value in superlattices whose quantum wells are strongly coupled. On the other hand, when the coupling is weak or when a strong electric field is applied to the superlattice, the noise becomes Poissonian. Although our results are qualitatively consistent with existing theories for one-dimensional multibarrier structures, the theories cannot account for the dependence of the noise on superlattice parameters that we have observed.