Magnetic fields may play an important role in the star-formation process, especially in the central regions of ＇starburst＇ galaxies where star formation is vigorous. But the field directions are very difficult to determine in the dense molecular gas out of which the stars form, so it has hitherto been impossible to test this hypothesis. Dust grains in interstellar clouds tend to be magnetically aligned, and it is possible to determine the alignment direction based on the polarization of optical light due to preferential extinction along the long axes of the aligned grains(1). This technique works, however, only for diffuse gas, not for the dense molecular gas. Here we report observations of polarized thermal emission from the aligned dust grains in the central region of M82, which directly traces(2) the magnetic field structure (as projected onto the plane of the sky). Organized field lines are seen around the brightest star-forming regions, while in the dusty halo the field lines form a giant magnetic bubble possibly blown out by the galaxy＇s ＇superwind＇.