We report on the observation of two neutrino-induced events which have an estimated deposited energy in the IceCube detector of 1.04 +/- 0.16 and 1.14 +/- 0.17 PeV, respectively, the highest neutrino energies observed so far. These events are consistent with fully contained particle showers induced by neutral-current nu(e,mu,tau) ((nu) over bar (e,mu,tau)) or charged-current nu(e) ((nu) over bar (e)) interactions within the IceCube detector. The events were discovered in a search for ultrahigh energy neutrinos using data corresponding to 615.9 days effective live time. The expected number of atmospheric background is 0.082 +/- 0.004(stat)(-0.057)(+0.041)(syst). The probability of observing two or more candidate events under the atmospheric background-only hypothesis is 2.9 x 10(-3) (2.8 sigma) taking into account the uncertainty on the expected number of background events. These two events could be a first indication of an astrophysical neutrino flux; the moderate significance, however, does not permit a definitive conclusion at this time.