The invention relates to a method for producing a poly-1-olefin by polymerisation of a 1-olefin of the formula R⁴CHCH₂, in which R⁴ is hydrogen or an alkyl radical having from 1 to 10 carbon atoms, in suspension, in solution or in the gas phase, at a temperature of from 20 to 200 C. and a pressure of from 0.5 to 50 bar, in the presence of a catalyst which consists of the product of the reaction of a magnesium alkoxide with a transition-metal compound (component a) and an organometallic compound (component b), whose component a has been produced by reacting a transition-metal compound of titanium, zirconium, vanadium or chromium with a gelatinous dispersion of the magnesium alkoxide in an inert hydrocarbon. In accordance with the invention, the gelatinous dispersion is obtained by stirring using a stirring element, or shearing using a high-performance shearing tool, a suspension of magnesium alkoxide powder having a mean particle size, expressed as the d₅₀ value, of less than 15 m, obtained by dry-grinding of commercially available magnesium alkoxide powder having a mean particle size, expressed as the d₅₀ value, of greater than or equal to 15 m, preferably in the range from 100 to 1000 m, in an inertized mill, in an aliphatic hydrocarbon in which the magnesium alkoxide particles are insoluble. The invention also relates to a catalyst which is suitable for carrying out the method.