Ultrafast time-resolved optical spectroscopy has revealed new classes of physical, chemical and biological reactions, in which directed, deterministic motions of atoms have a key role. This contrasts with the random, diffusive motion of atoms across activation barriers that typically determines kinetic rates on slower timescales. An example of these new processes is the ultrafast melting of semiconductors, which is believed to arise from a strong modification of the inter-atomic forces owing to laser-induced promotion of a large fraction (10% or more) of the valence electrons to the conduction band. The atoms immediately begin to move and rapidly gain sufficient kinetic energy to induce melting--much faster than the several picoseconds required to convert the electronic energy into thermal motions. Here we present measurements of the characteristic melting time of InSb with a recently developed technique of ultrafast time-resolved X-ray diffraction that, in contrast to optical spectroscopy, provides a direct probe of the changing atomic structure. The data establish unambiguously a loss of long-range order up to 900 A inside the crystal, with time constants as short as 350 femtoseconds. This ability to obtain the quantitative structural characterization of non-thermal processes should find widespread application in the study of ultrafast dynamics in other physical, chemical and biological systems.