Singularities in classical general relativity are unavoidable, but perhaps only represent the fact that we are extrapolating our theory beyond its region of validity. We study the end stages of gravitational collapse using the functional Schrodinger formalism to capture quantum effects in the near singularity limit. We find that that the equations of motion which govern the behavior of the collapsing object near the classical singularity become strongly non-local. We managed to solve the non-local equations, and found an explicit form of the wavefunction describing the collapsing object. This wavefunction and the corresponding probability density are non-singular at the origin, thus indicating that quantization should be able to rid gravity of singularities, just as it was the case with the singular Coulomb potential.