A search for the relic neutrinos from all past core-collapse supernovae was conducted using 1496 days of data from the Super-Kamiokande detector. This analysis looked for electron-type antineutrinos that had produced a positron with an energy greater than 18 MeV. In the absence of a signal, 90% C.L. upper limits on the total flux were set for several theoretical models; these limits ranged from 20 to 130 (nu) over bar (e) cm(-2) s(-1). Additionally, an upper bound of 1.2 (nu) over bar (e) cm(-2) s(-1) was set for the supernova relic neutrino flux in the energy region E-nu >19.3 MeV.