  We define weakly proper Fell bundles and construct exotic fixed point
algebras for such bundles. Three alternative constructions of such algebras are
given. Under a kind of freeness condition, one of our constructions implies
that every exotic cross sectional C*-algebra of a weakly proper Fell bundle is
Morita equivalent to an exotic fixed point algebras. The other constructions
are used to show that ours generalizes that of Buss and Echterhoff on weakly
proper actions on C*-algebras. We also generalize to Fell bundles the fact that
every C*-action which is proper in Kasparov's sense is amenable.
