We examine the evolution of a 3-brane that separates two regions of a4+1dimensional bulk space-time with potentially different cosmological constants. We have included a term in the brane action for the scalar curvature. When one or both of the regions has a zero cosmological constant, we find solutions for which the cosmology on the brane has the same time-dependence as a standard radiation-dominated universe, but which depends on a mass parameter in the bulk metric rather than the radiation density on the brane. When a non-zero cosmological constant is present in both bulk regions, it is possible to obtain a standard Robertson-Walker evolution. We then extract the effective theory of gravity in the weak-field limit seen by an observer on the brane in each of these scenarios. ©2000 American Institute of Physics.