In this article, we consider a bouncing Universe, as described for example by Loop Quantum Cosmology. If the current acceleration is due to a true cosmological constant, this constant is naturally conserved through the bounce and the Universe should also be in a (contracting) de Sitter phase in the remote past. We investigate here the possibility that the de Sitter temperature in the contracting branch fills the Universe with radiation and causes the bounce and the subsequent inflation and reheating. We also consider the possibility that this gives rise to a cyclic model of the Universe and suggest some possible tests.