As was pointed out, the question did not intend the set enclosed by to be the interior of the curve. The question now boils down to whether a differentiable Jordan curve is openhas measure zero. No smoothness This follows from the Lebesgue density theorem, see e.g. http://en.wikipedia.org/wiki/Lebesgue's_density_theorem.
If the curve is differentiable, then at each point of the curve the Lebesgue density is neededzero. By the Lebesgue density theorem, the measure of the curve must be zero.