We consider a cosmological model in which an unstable massive relic particle species (denoted by X) has an initial mass density relative to baryons .beta./sup -1/ identically equal rho/sub X//rho/sub B/ >> 1, and then decays recently (redshift z less than or equal to 1000) into particles which are still relativistic today (denoted by R). We write down and solve the coupled equations for the cosmic scale factor a(t), the energy density in the various components (rho/sub X/, rho/sub R/, rho/sub B/), and the growth of linear density perturbations (delta rho/rho). The solutions form a one parameter (.beta.) family of solutions; physically .beta./sup -1/ approx. = (.cap omega./sub R//.cap omega./sub NR/) x (1 + z/sub D/) = (ratio today of energy density of relativistic to nonrelativistic particles) x (1 + redshift of (decay)). We discuss the observational implications of such a cosmological model and compare our results to earlier results computed in the simultaneous decay approximation. In an appendix we briefly consider the case where one of the decay products of the X is massive and becomes nonrelativistic by the present epoch. 21 references.