Primordial Black Holes (PBHs) may have been created by density fluctuations in the early Universe and could be as massive as $> 10^9$ solar masses or as small as the Planck mass. It has been postulated that a black hole has a temperature inversely-proportional to its mass and will thermally emit all species of fundamental particles via Hawking Radiation. PBHs with initial masses of $sim 5 times 10^{14}$ g (approximately one gigaton) should be expiring today with bursts of high-energy gamma radiation in the GeV--TeV energy range. The High Altitude Water Cherenkov (HAWC) Observatory is sensitive to gamma rays with energies of $sim$300 GeV to past 100 TeV, which corresponds to the high end of the PBH burst spectrum. With its large instantaneous field-of-view of $sim 2$ sr and a duty cycle over 95%, the HAWC Observatory is well suited to perform an all-sky search for PBH bursts. We conducted a search using 959 days of HAWC data and exclude the local PBH burst rate density above $3400~mathrm{pc^{-3}~yr^{-1}}$ at 99% confidence, the strongest limit on the local PBH burst rate density from any existing electromagnetic measurement.