We prove that the image of an isometric embedding into R3 of a two dimensional complete Riemannian manifold (Σ,g) without boundary is a convex surface, provided that, first, both the embedding and the metric g enjoy a C1,α regularity for some α>2/3, and second, the distributional Gaussian curvature of g is nonnegative and nonzero. The analysis must pass through some key observations regarding solutions to the very weak Monge-Ampère equation.