We study limits of quasi-Fuchsian 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 [24, Conjecture 6.5] that the closures of
pleating varieties for quasi-Fuchsian 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 $2\pi$ .