Deciphering the three-dimensional (3D) structure of complex molecules is of major importance, typically accomplished with X-ray crystallography. Unfortunately, many important molecules cannot be crystallized, hence their 3D structure is unknown. Ankylography presents an alternative, relying on scattering an ultrashort X-ray pulse off a single molecule before it disintegrates, measuring the far-field intensity on a two-dimensional surface, followed by computation. However, significant information is absent due to lower dimensionality of the measurements and the inability to measure the phase. Recent Ankylography experiments attracted much interest, but it was counter-argued that Ankylography is valid only for objects containing a small number of volume pixels. Here, we propose a sparsity-based approach to reconstruct the 3D structure of molecules. Sparsity is natural for Ankylography, because molecules can be represented compactly in stoichiometric basis. Utilizing sparsity, we surpass current limits on recoverable information by orders of magnitude, paving the way for deciphering the 3D structure of macromolecules.