Conditions are investigated under which a proper leaf has a trivally foliated, saturated neighborhood. We consider transversely orientable foliations of closed, orientable 3-manifolds by (necessarily orientable) surfaces. These hypotheses will be assumed throughout, save mention to the contrary. Definition. A surface L has the C'-stability property if, whenever L is homeomorphic to a proper leaf of a Cr foliation S~, that leaf admits an open, ^-saturated neighborhood U, fibered over Sx by ^\U. C "-stability property [Im, p. 622].