The astrophysical sources of the extraterrestrial, very high-energy neutrinos detected by the IceCube collaboration remain to be identified. Gamma-ray (γ-ray) blazars have been predicted to yield a cumulative neutrino signal exceeding the atmospheric background above energies of 100 TeV, assuming that both the neutrinos and the γ-ray photons are produced by accelerated protons in relativistic jets. As the background spectrum falls steeply with increasing energy, the individual events with the clearest signature of being of extraterrestrial origin are those at petaelectronvolt energies. Inside the large positional-uncertainty fields of the first two petaelectronvolt neutrinos detected by IceCube, the integrated emission of the blazar population has a sufficiently high electromagnetic flux to explain the detected IceCube events, but fluences of individual objects are too low to make an unambiguous source association. Here, we report that a major outburst of the blazar PKS B1424-418 occurred in temporal and positional coincidence with a third petaelectronvolt-energy neutrino event (HESE-35) detected by IceCube. On the basis of an analysis of the full sample of γ-ray blazars in the HESE-35 field, we show that the long-term average γ-ray emission of blazars as a class is in agreement with both the measured all-sky flux of petaelectronvolt neutrinos and the spectral slope of the IceCube signal. The outburst of PKS B1424-418 provides an energy output high enough to explain the observed petaelectronvolt event, suggestive of a direct physical association.
