We report the e INTernational Gamma-ray Astrophysics Laboratory (INTEGRAL) detection of the short gamma-ray burst GRB 170817A (discovered by Fermi-GBM) with a signal-to-noise ratio of 4.6, and, for the first time, its association with the gravitational waves (GWs) from binary neutron star (BNS) merging event GW170817 detected by the LIGO and Virgo observatories. The significance of association between the gamma-ray burst observed by INTEGRAL and GW170817 is 3.2 $sigma$, while the association between the Fermi-GBM and INTEGRAL detections is 4.2 $sigma$. GRB 170817A was detected by the SPI-ACS instrument about 2 s after the end of the gravitational wave event. We measure a fluence of $(1.4 pm 0.4 pm 0.6) times$10$^{-7}$ erg cm$^{-2})$ (75--2000 keV), where, respectively, the statistical error is given at the 1 $sigma$ confidence level, and the systematic error corresponds to the uncertainty in the spectral model and instrument response. We also report on the pointed follow-up observations carried out by INTEGRAL, starting 19.5 h after the event, and lasting for 5.4 days. We provide a stringent upper limit on any electromagnetic signal in a very broad energy range, from 3 keV to 8 MeV, constraining the soft gamma-ray afterglow flux to $<7.1times$10$^{-11}$ erg cm$^{-2}$ s$^{-1}$ (80--300 keV). Exploiting the unique capabilities of INTEGRAL, we constrained the gamma-ray line emission from radioactive decays that are expected to be the principal source of the energy behind a kilonova event following a BNS coalescence. Finally, we put a stringent upper limit on any delayed bursting activity, for example from a newly formed magnetar.
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