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-rstability property if, whenever L is homeomorphic to a proper leaf of a Cr foliation C, that leaf admits an open, '5-saturated neighborhood U, fibered over S 1 by C1 U. DEFINITION. A surface L has the weak Cr-stability property if, whenever L is homeomorphic to a proper leaf of a Cr foliation CY, such that the germinal holonomy group of the leaf is trivial, then that leaf admits an open, C-saturated neighborhood U, fibered over R by 15I U. In [In], it was shown (essentially) that the plane has the C '-stability property and it was announced that every surface with finitely generated fundamental group has the weak C '-stability property. This condition on the fundamental group implies that the surface has only finitely many ends and finitely many handles (for the concept of see [A-S], [Ri], [C-Cl], etc.). For C2 foliations, these results generalize considerably. THEOREM 1. Every orientable surface with at most countably many ends and finitely many handles has the weak C2-stability property. THEOREM 2. Every orientable surface with at most countably many ends and no handles has the C 2-stability property. REMARKS. (1) By the constructive proof of [C-Cl, Theorem 2], the condition in Theorem 2 does not imply the Cl-stability property. In fact, the cylinder is not C '-stable. (2) It is easy to construct an example showing that the surface with two ends, both nonplanar (hence with infinitely many handles), does not have the weak C ?-stability property [Im, p. 622]. Received by the editors October 2, 1980. 1980 Mathematics Subject Classification. Primary 57R30. 'Partially supported by NSF Grant MCS-8001547. 2Partially supported by NSF Grant MCS-8003248. ? 1981 American Mathematical Society 0002-9939/81 /0000-0490/$01.75