The IceCube Neutrino Observatory has reported the observation of 35 neutrino events above 30 TeV with evidence for an astrophysical neutrino flux using data collected from May 2010 to May 2013. These events provide the first high-energy astrophysical neutrino flux ever observed.The sources of these events are currently unknown. IceCube has looked for correlations between these events and a list of TeV photon sources including a catalog of 36 galactic sources and 42 extragalactic sources, correlations with the galactic plane and center, and spatial and temporal clustering.These searches have shown no significant correlations.The isotropic distribution of the event directions gives indications that the events could be extragalactic in nature and therefore may originate in the same processes that generate ultra-high-energy cosmic rays (UHECRs).The sources of these UHECRs are still unknown; however, gamma-ray bursts (GRBs) have been proposed as one possible source class.By determining the source of these high-energy neutrinos, it may be possible to determine the sources of UHECRs as well.This study is a search for directional and temporal correlation between 856 GRBs and the astrophysical neutrino flux observed by IceCube.Nearly 10,000 expanding time windows centered on the earliest reported time of the burst were examined.The time windows start at ±10 s and extend to ±15 days.We find no evidence of correlations for these time windows and set an upper limit on the fraction of the astrophysical flux that can be attributed to the observed GRBs as a function of the time window.GRBs can contribute at most 12% of the astrophysical neutrino flux if the neutrino-GRB correlation time is less than ≈20 hours, and no more than 38% of the astrophysical neutrino flux can be attributed to the known GRBs at time scales up to 15 days. We conclude that GRBs observable by satellites are not solely responsible for IceCube’s astrophysical neutrino flux, even if very long correlation time scales are assumed.