The existence of Dactyl, the small satellite of asteroid 243 Ida, presents an intriguing paradox: if exposed to the same projectile bombardment as Ida, it should have been disrupted long ago. To solve this paradox, it has been proposed that either Ida (and the entire Koronis family) is relatively young ($\approx 100$~Myr), or Dactyl has reaccreted many times from its own debris after having been disrupted. Here we propose a third alternative, that is that Dactyl is much younger than Ida, and it was formed by rotational bursting of a precursor fragment ejected from Ida after an impact. We discuss some recent experimental results showing that sizable fragments from shattered targets do undergo rotational bursting and are fissioned after traveling over a length of several target diameters; the relative speed between the fission remnants is comparable to the initial ejection velocity. Then we have performed a number of numerical integrations of the orbits of fictitious particles resulting from an assumed rotational bursting event in the gravitational field of Ida; the results show that, depending on the initial conditions, up to several percent of the particles get trapped into stable satellite--like orbits resembling that of Dactyl. We conclude that this mechanism may have formed Dactyl in the last $\approx 10\%$ of Ida's lifetime, either after an energetic cratering impact or (more probably) after a collision which shattered Ida without dispersing most of its fragments ``to infinity''.