Linear defects are generic in continuous media. In quantum systems they appear as topological line defects which are associated with a circulating persistent current. In relativistic quantum field theories they are known as cosmic strings, in superconductors as quantized flux lines, and in superfluids and low-density Bose-Einstein condensates as quantized vortex lines. A conventional quantized vortex line consists of a central core around which the phase of the order parameter winds by 2πn, while within the core the order parameter vanishes or is depleted from the bulk value. Usually vortices are singly quantized (that is, have <italic>n</italic> = 1). But it has been theoretically predicted that, in superfluid 3He-A, vortex lines are possible that have <italic>n</italic> = 2 and continuous structure, so that the orientation of the multi-component order parameter changes smoothly throughout the vortex while the amplitude remains constant. Here we report direct proof, based on high-resolution nuclear magnetic resonance measurements, that the most common vortex line in 3He-A has <italic> n</italic> = 2. One vortex line after another is observed to form in a regular periodic process, similar to a phase-slip in the Josephson effect.