  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 $0.08\, Gpc^{-3}yr^{-1}$.
