  We study limits of quasifuchsian groups for which the bending measures on the
convex hull boundary tend to zero, giving necessary and sufficient conditions
for the limit group to exist and be Fuchsian. As an application we complete the
proof of a conjecture made in \cite{S1}, that the closure of pleating varieties
for quasifuchsian groups meet Fuchsian space exactly in Kerckhoff's lines of
minima of length functions. Doubling our examples gives rise to a large class
of cone manifolds which degenerate to hyperbolic surfaces as the cone angles
approach $2\pi$.
