We report an unexpected characteristic of dislocation cores in silicon. Using first-principles calculations, we show that all of the stable core configurations for a nondissociated 60 degrees dislocation are sessile. The only glissile configuration, previously obtained by nucleation from surfaces, surprisingly corresponds to an unstable core. As a result, the 60 degrees dislocation motion is solely driven by stress, with no thermal activation. We predict that this original feature could be relevant in situations for which large stresses occur, such as mechanical deformation at room temperature. Our work also suggests that postmortem observations of stable dislocations could be misleading and that mobile unstable dislocation cores should be taken into account in theoretical investigations.