  We study the spectral stability of a family of periodic wave trains of the
Korteweg-de Vries/Kuramoto-Sivashinsky equation $ \partial_t v+v\partial_x
v+\partial_x^3 v+\delta(\partial_x^2 v +\partial_x^4 v)=0$, $\delta>0$, in the
Korteweg-de Vries limit $\delta\to 0$, a canonical limit describing
small-amplitude weakly unstable thin film flow. More precisely, we carry out a
rigorous singular perturbation analysis reducing the problem to the evaluation
for each Bloch parameter $\xi\in [0,2\pi]$ of certain elliptic integrals
derived formally (on an incomplete set of frequencies/Bloch parameters, hence
as necessary conditions for stability) and numerically evaluated by Bar and
Nepomnyashchy \cite{BN}, thus obtaining, up to machine error, complete
conclusions about stability. The main technical difficulty is in treating the
large-frequency and small Bloch-parameter regimes not studied by Bar and
Nepomnyashchy \cite{BN}, which requires techniques rather different from
classical Fenichel-type analysis. The passage from small-$\delta$ to
small-$\xi$ behavior is particularly interesting, using in an essential way an
analogy with hyperbolic relaxation at the level of the Whitham modulation
equations.
