  We show that in a large class of two dimensional models with conformal matter
fields, the semiclassical cosmological solutions have a weak coupling
singularity if the classical matter content is below a certain threshold. This
threshold and the approach to the singularity are model-independent. When the
matter fields are not conformally invariant, the singularity persists if the
quantum state is the vacuum near the singularity, and could dissappear for
other quantum states.
