An unfolding procedure is described whereby the known electron energy distribution function in the swarm F(( epsilon ), epsilon ) at each mean electron energy ( epsilon ) is used as the 'smearing out' function to unfold the monoenergetic attachment rate M( epsilon ) from the experimentally determined electron attachment rate functions R_exp(( epsilon )). From the monoenergetic attachment rate the absolute electron attachment cross section as a function of electron energy epsilon is calculated. The cross section functions determined from the unfolding of swarm data for a number of molecules compare well with those obtained using quasi-monoenergetic electron beams. This successful unfolding of swarm data enhances the potential of the electron swarm method in electron attachment studies.