There is a canonical identification, due independently to the author and to F. Labourie, of a convex real projective structure on an orientable surface of genus g and a pair consisting of a conformal structure together with a holomorphic cubic differential on the surface. The Deligne–Mumford compactification of the moduli space of curves then suggests a partial compactification of the moduli space of convex real projective structures: Allow the Riemann surface to degenerate to a stable nodal curve on which there is a regular cubic differential. We construct convex real projective structures on open surfaces corresponding to this singular data and relate their holonomy to earlier work of Goldman. Also we have results for families degenerating toward the boundary of the moduli space. The techniques involve affine differential geometry results of Cheng–Yau and C.P. Wang and a result of Dunkel on the asymptotics of systems of ODEs.