Results are reported from a search for low-multiplicity neutrino bursts in the Sudbury Neutrino Observatory. Such bursts could indicate the detection of a nearby core-collapse supernova explosion. The data were taken from Phase I (1999 November-2001 May), when the detector was filled with heavy water, and Phase II (2001 July-2003 August), when NaCl was added to the target. The search was a blind analysis in which the potential backgrounds were estimated and analysis cuts were developed to eliminate such backgrounds with 90% confidence before the data were examined. The search maintained a greater than 50% detection probability for standard supernovae occurring at a distance of up to 60 kpc for Phase I and up to 70 kpc for Phase II. No low-multiplicity bursts were observed during the data-taking period.