The core of the nearby and very concentrated globular cluster NGC 6397 has been imaged through the f/96-F140W, f/48-F140W, f/96-F210M and f/48-F220W ultraviolet filters of the Faint Object Camera on board the Hubble Space Telescope down to an ultraviolet limiting magnitude of ~19. The most interesting objects in the field of view are six very bright centrally concentrated, blue stragglers observed for the first time in the UV. Using these and other data from ground-based observations, we have been able to deduce from a comparison with Kurucz's atmosphere models, temperatures of ~10000K and masses of ~1.6M_{sun}_ for the four brightest ones, which is remarkably close to twice the turn-off mass of NGC 6397. This finding supports the idea that two-star mechanisms (collisions, mergers) are at the origin of the blue stragglers in the core of NGC 6397. Since the central density is very high, collisions between main sequence stars are frequent, therefore providing the best formation mechanism. We have computed the number of such collisions in the core of NGC 6397 and found it to be of the same order as the number of bright blue stragglers observed there. Thanks to the HST resolution, we have also been able to resolve one object, previously classified as a yellow straggler, into a blend of a blue straggler and three redder stars.