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 (PSD) down to an electron-equivalent energy of 20 keV. In the surface dataset using a triple-coincidence tag we found the fraction of beta events that are misidentified as nuclear recoils to be $<1.4times 10^{-7}$ (90% C.L.) for energies between 43-86 keVee 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 with only a double-coincidence tag. The combined data set contains $1.23times10^8$ events. One of those, in the underground data set, is in the nuclear-recoil region of interest. Taking into account the expected background of 0.48 events coming from random pileup, the resulting upper limit on the electronic recoil contamination is $<2.7times10^{-8}$ (90% C.L.) between 44-89 keVee 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 PSD 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 approx. $10^{-10}$ for an electron-equivalent energy threshold of 15 keV for a detector with 8 PE/keVee 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.
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