Attraction between the atoms of a Bose-Einstein condensate renders it unstable to collapse, although a condensate with a limited number of atoms [1] can be stabilized [2] by confinement in an atom trap. However, beyond this number the condensate collapses [3, 4, 5]. Condensates constrained to one-dimensional motion with attractive interactions are predicted to form stable solitons, in which the attractive forces exactly compensate for wave-packet dispersion [1]. Here we report the formation of bright solitons of 7Li atoms in a quasi-one-dimensional optical trap, by magnetically tuning the interactions in a stable Bose-Einstein condensate from repulsive to attractive. The solitons are set in motion by offsetting the optical potential, and are observed to propagate in the potential for many oscillatory cycles without spreading. We observe a soliton train, containing many solitons; repulsive interactions between neighbouring solitons are inferred from their motion.