The general solution to the semiclassical backreaction equation is found for conformally invariant free quantum fields in spatially flat homogeneous and isotropic spacetime with Cosmological constant and with no classical source when the ratio of the renormalisation parametersβ/α=9/4. It contains a two-parameter family of bouncing solutions that avoid the singularity. There are several one-parameter families which do not have particle horizons. The stability of these solutions is investigated and it is found that they are stable whenα andλ have different signs. However, when both parameters have the same sign the set of stable solutions is restricted by the condition 0 <λ < 1/9. In both cases these solutions have a final de Sitter stage.