The giant planets of the Solar System formed on initially compact orbits, which transitioned to the current wider configuration by means of an orbital instability. The timing of that instability is poorly constrained. In this work, we use dynamical simulations to demonstrate that the instability implanted planetesimal fragments from the terrestrial planet region into the asteroid main belt. We use meteorite data to show that the implantation occurred >60 million years (Myr) after the Solar System began to form. Combining this constraint with a previous upper limit derived from Jupiter's trojan asteroids, we conclude that the orbital instability occurred 60 to 100 Myr after the beginning of Solar System formation. The giant impact that formed the Moon occurred within this range, so it might be related to the giant planet instability.