In search of non-inflationary cosmological models that do not suffer from the horizon and the entropy problems of the standard hot big-bang model, we present two exact cosmological solutions to Einstein's equations based on a homogeneous scaler field with an exponential potential. The curvature constant k is -1 and 0 in these solutions. The model universe of these solutions is singular, causally connected, entropy-conserving and devoid of matter. We extend these solutions to the more realistic case of a universe in which radiation and nonrelativistic matter are created through the decay of the homogeneous scalar field. The energy density of the scalar held as well as those of the radiation and non-relativistic matter vary with t(-2). The initial entropy of the universe is zero and created as the scalar field decays into radiation. The t(-2) variation of the energy densities guarantees a successful cosmic helium synthesis which is essentially indistinguishable from that in the standard hot big-bang model. We discuss other aspects of these models and argue that they are viable singular models, free of some of the problems of the standard model.