Observations of the radio remnant of supernova 1987A using the Australia Telescope Compact Array in the 12 mm band on 2003 July 31 (day 6002.7 after the explosion) give the first fully resolved radio image of the supernova remnant. The diffraction-limited image has a resolution of about 0farcs45, a factor of 2 better than that of the previously obtained 3 cm images. There is excellent agreement between the 12 mm image and a contemporaneous superresolved 3 cm image. Superresolution of the 12 mm image gives a further factor of 2 improvement in resolution, to 0farcs25, albeit with limited dynamic range. While the spatial distributions of the radio and X-ray emission are broadly similar, there are significant differences in detail, with no correspondence in the regions of brightest emission. The 12 mm image is well modeled by a thick equatorial ring inclined at 43° to the line of sight. This, together with the common east-west asymmetry and the relatively steady increase in the radio flux density, suggests that the reverse shock is the main site for generation of the radio emission.