We demonstrate experimentally an efficient electron axicon lens using a magnetic vortex. We show that naturally occurring magnetic vortices with circular magnetic moment distributions in a soft-magnetic thin film create conical phase shifts for fast electrons. Such radially symmetric linear phase ramps are equivalent to ideal light optical axicons. We apply this lens to generate efficient nondiffracting electron Bessel beams, which we observe experimentally in through-focus Lorentz images as well as in propagated off-axis electron holograms. This highlights the potential for using magnetic nanostructures as highly efficient and flexible phase plates for crafting desired electron beam shapes.