[en] The γ-ray burst GRB970508 was observed simultaneously by the Gamma Ray Burst Monitor (GRBM) and one of the X-ray Wide Field Cameras (WFC) aboard BeppoSAX. The latter provided a position within 1.9' radius. A series of follow-up observations with the Narrow Field Instruments (NFI) was then performed in a period from ∼ 6 hours to 6 days after the main event. A previously unknown source, which we associate with the afterglow of the GRB, was discovered in the error box. We find that, after the initial burst, X-ray emission is still present and decays as ∼ t^-1.1 up to ∼ 6 x 10⁴ s. This is followed by a burst of activity with a duration ∼ 10⁵ s. The energy produced in this event is a substantial fraction of the total energy of the GRB, which means that the afterglow is not a remnant of the initial burst (the GRB) that fades away smoothly. Our results support the idea that the processes generating the GRB and its afterglow are the same