  We prove that the image of an isometric embedding into ${\mathbb R}^3$ of a
two dimensionnal complete Riemannian manifold $(\Sigma, g)$ without boundary is
a convex surface provided both the embedding and the metric $g$ enjoy a
$C^{1,\alpha}$ regularity for some $\alpha>2/3$ and 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-Amp\`ere
equation.
