Planetary nebulae are thought to be formed when a slow wind from the progenitor giant star is overtaken by a subsequent fast wind generated as the star enters its white dwarf stage(1). A shock forms near the boundary between the winds, creating the relatively dense shell characteristic of a planetary nebula. A spherically symmetric wind will produce a spherically symmetric shell, yet over half of known planetary nebulae are not spherical; rather, they are elliptical or bipolar in shape(2). A magnetic field could launch and collimate a bipolar outflow, but the origin of such a field has hitherto been unclear, and some previous work has even suggested that a field could not be generated(3). Here we show that an asymptotic-giant-branch (AGB) star can indeed generate a strong magnetic field, having as its origin a dynamo at the interface between the rapidly rotating core and the more slowly rotating envelope of the star. The fields are strong enough to shape the bipolar outflows that produce the observed bipolar planetary nebulae. Magnetic braking of the stellar core during this process may also explain the puzzlingly(4) slow rotation of most white dwarf stars.