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Take a set $K\subset\mathbb{R}^2$ that is homeomorphic to the Cantor set and has positive $2$-dimensional Lebesgue measure. Take a homeomorphism of the ternary Canor set onto $K$ and extend it as a piecewise $C^\infty$ curves in the complement of the ternary Cantor set. If you make it carefully, you will obtain an injective curve whose image has positive 2-dimensional Lebesgue measure and which is $C^\infty$ smooth in the complement of the set of measure zero - in the complement of the ternary Cantor set.