CLASSICAL relativistic singularities remain one of the most puzzling problems of contemporary physics. Although no consensus of opinion has been attained on their status, a great deal of work has been devoted to the search for physical mechanisms capable of eliminating the cosmological singularities, whose existence follows from the theorems of Hawking and Penrose1,2. The alternative view that the initial singularity of big-bang models is essential for the coming into being of the Universe, has also been advocated3,4. Some suggestions of cosmological singularity avoidance have recently come to light, in the `classical' quantum regime (from 10-43-10-23s) (refs 5-8) or even without resorting to quantum effects9. The quantum gravitational period (t < 10-43 s) remains potentially very interesting for searching for singularity avoidance mechanisms. An interesting approach is that of quantum cosmology, based on the hamiltonian formulation of general relativity (Dirac's10 or Arnowitt, Deser and Misner's11 (ADM) method). This approach was initiated by De Witt12 and then followed by Misner13 and many others (see refs 14,15) and is used here to derive a non-singular quantum cosmological model.