The formation of molecular clouds, which serve as stellar nurseries in galaxies, is poorly understood. A class of cloud formation models suggests that a large-scale galactic magnetic field is irrelevant at the scale of individual clouds, because the turbulence and rotation of a cloud may randomize the orientation of its magnetic field(1,2). Alternatively, galactic fields could be strong enough to impose their direction upon individual clouds(3,4), thereby regulating cloud accumulation and fragmentation(5), and affecting the rate and efficiency of star formation(6). Our location in the disk of the Galaxy makes an assessment of the situation difficult. Here we report observations of the magnetic field orientation of six giant molecular cloud complexes in the nearby, almost face-on, galaxy M33. The fields are aligned with the spiral arms, suggesting that the large-scale field in M33 anchors the clouds.