We report the results of the first complete survey of an area of the Galactic plane for maser emission from the 6.7-GHz 51 → 60 A+ transition of CH3OH. The survey covers a 10.6-deg2 region of the Galactic plane in the longitude range 325°-335° and latitude range -0°.53-0°.53. The survey is sensitive to masers with a peak flux density greater than ∼ 2.6 Jy. The weakest maser detected has a peak flux density of 2.3 Jy, and the strongest a peak flux density of 425 Jy. We detected a total of 50 distinct masers, 26 of which are new detections. We show that many 6.7-GHz CH3OH masers are not associated with IRAS sources, and that some are associated with sources that have colours differing from those of a typical ultracompact H II (UCH II) region. We estimate that the number of UCH II regions in the Galaxy is significantly more than suggested by IRAS-based estimates, possibly by more than a factor of 2.