The DEAP-1 low-background liquid argon detector was used to measure scintillation pulse shapes of electron and nuclear recoil events and to demonstrate the feasibility of pulse-shape discrimination down to an electron-equivalent energy of 20 keV ee .In the surface dataset using a triple-coincidence tag we found the fraction of β events that are misidentified as nuclear recoils to be < 1.4 × 10 −7 (90% C.L.) for energies between 43-86 keV ee and for a nuclear recoil acceptance of at least 90%, with 4% systematic uncertainty on the absolute energy scale. The discrimination measurement on surface was limited by nuclear recoils induced by cosmic-ray generated neutrons. This was improved by moving the detector to the SNOLAB underground laboratory, where the reduced background rate allowed the same measurement to be done with only a double-coincidence tag.The combined data set contains 1.23 × 10 8 events. One of those, in the underground data set, is in the nuclearrecoil region of interest. Taking into account the expected background of 0.48 events coming from random pileup, the resulting upper limit on the level of electronic recoil contamination is < 2.7 × 10 −8 (90% C.L.) between 44-89 keV ee and for a nuclear recoil acceptance of at least 90%, with 6% systematic uncertainty on the absolute energy scale.We developed a general mathematical framework to describe pulse-shape-discrimination parameter distributions and used it to build an analytical model of the distributions observed in DEAP-1. Using this model, we project a misidentification fraction of approximately 10 −10 for an electron-equivalent energy threshold of 15 keV ee for a detector with 8 PE/keV ee light yield. This reduction enables a search for spin-independent scattering of WIMPs from 1000 kg of liquid argon with a WIMP-nucleon cross-section sensitivity of 10 −46 cm 2 , assuming negligible contribution from nuclear recoil backgrounds.