We present new imaging observations of the remnant of Supernova (SN) 1987A at 44 GHz, performed in 2011 with the Australia Telescope Compact Array (ATCA). The $0\farcs35\times0\farcs23$ resolution of the diffraction-limited image is the highest achieved to date in high-dynamic range. We also present a new ATCA image at 18 GHz derived from 2011 observations, which is super-resolved to $0\farcs25$. The flux density is 40$\pm$2 mJy at 44 GHz and 81$\pm$6 mJy at 18 GHz. At both frequencies, the remn...