  We show that any isometric immersion of a flat plane domain into $\mathbb
R^3$ is developable provided it enjoys the little H\"older regulairty
$c^{1,2/3}$. In particular, isometric immersions of local $C^{1,\alpha}$
regularity with $\alpha > 2/3$ belong to this class. The proof is based on the
existence of a weak notion of second fundamental form for such immersions, the
analysis of the Gauss-Codazzi-Mainardi equations in this weak setting, and a
parallel result on the very weak solutions to the degenerate Monge-Amp\`ere
equation analyzed by Lewicka and the second author.
