Disclosed is a method for producing polyisobutene containing at least 75 mole percent of terminal vinylidene groupings. According to the inventive method, isobutene or hydrocarbon mixtures containing isobutene is/are polymerized in a liquid phase in the presence of a boron trifluoride complex catalyst having the composition a(BF3): b(Co1): c(Co2), wherein Co1 represents at least one tertiary alcohol, Co2 represents a compound selected among water, primary alcohols, secondary alcohol, dialkyl ethers, alkanecarboxylic acids, and phenols, the ratio c:b ranges from 0.9 to 1.8, and the ratio (b+c):a ranges from 0.9 to 3.0.