Intermediate-mass black holes (IMBHs) span the approximate mass range $100$--$10^5,M_odot$, between black holes (BHs) formed by stellar collapse and the supermassive BHs at the centers of galaxies. Mergers of IMBH binaries are the most energetic gravitational-wave sources accessible by the terrestrial detector network. Searches of the first two observing runs of Advanced LIGO and Advanced Virgo did not yield any significant IMBH binary signals. In the third observing run (O3), the increased network sensitivity enabled the detection of GW190521, a signal consistent with a binary merger of mass $sim 150,M_odot,$ providing direct evidence of IMBH formation. Here we report on a dedicated search of O3 data for further IMBH binary mergers, combining both modelled (matched filter) and model independent search methods. We find some marginal candidates, but none are sufficiently significant to indicate detection of further IMBH mergers. We quantify the sensitivity of the individual search methods and of the combined search using a suite of IMBH binary signals obtained via numerical relativity, including the effects of spins misaligned with the binary orbital axis, and present the resulting upper limits on astrophysical merger rates. Our most stringent limit is for equal mass and aligned spin BH binary of total mass $200,M_odot$ and effective aligned spin 0.8 at $0.056,Gpc^{-3} yr^{-1}$ (90 $%$ confidence), a factor of 3.5 more constraining than previous LIGO-Virgo limits. We also update the estimated rate of mergers similar to GW190521 to $0.08, Gpc^{-3}yr^{-1}$.
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