Primordial Black Holes (PBHs) are hypothetical black holes predicted to have been formed from density fluctuations in the early Universe. PBHs with an initial mass around $10^{14}-10^{15}$g are expected to end their evaporation at present times in a burst of particles and very-high-energy (VHE) gamma rays. Those gamma rays may be detectable by the High Energy Stereoscopic System (H.E.S.S.), an array of imaging atmospheric Cherenkov telescopes. This paper reports on the search for evaporation bursts of VHE gamma rays with H.E.S.S., ranging from 10 to 120 seconds, as expected from the final stage of PBH evaporation and using a total of 4816 hours of observations. The most constraining upper limit on the burst rate of local PBHs is $2000$ pc$^{-3}$ yr$^{-1}$ for a burst interval of 120 seconds, at the 95\% confidence level. The implication of these measurements for PBH dark matter are also discussed.
