  We study the application of recursion relations to the calculation of finite
one-loop gravity amplitudes. It is shown explicitly that the known four, five,
and six graviton one-loop amplitudes for which the external legs have identical
outgoing helicities, and the four graviton amplitude with helicities (-,+,+,+)
can be derived from simple recursion relations. The latter amplitude is derived
by introducing a one-loop three-point vertex of gravitons of positive helicity,
which is the counterpart in gravity of the one-loop three-plus vertex in
Yang-Mills. We show that new issues arise for the five point amplitude with
helicities (-,+,+,+,+), where the application of known methods does not appear
to work, and we discuss possible resolutions.
