Towards a finite quantum gravity

It is shown that in Einstein quantum gravity one can achieve the cancellation of all the ultraviolet divergences on a mass shell by an appropriate choice of the background metric. In the framework of dimensional regularization g sub mu nu sup B = sigma sub larger than or = 0 g sub mu nu sup (k)x epsilon sup k, where the classical metric g/sub mu nu sup (0)obeys the Einstein equation R sub mu nu(g sup (0))=0.