ABSTRACT We have modelled the near-infrared to radio images of the Crab Nebula with a Bayesian SED model to simultaneously fit its synchrotron, interstellar (IS), and supernova dust emission. We infer an IS dust extinction map with an average AV = 1.08 ± 0.38 mag, consistent with a small contribution (${\lesssim }22{{\ \rm per\ cent}}$) to the Crab’s overall infrared emission. The Crab’s supernova dust mass is estimated to be between 0.032 and 0.049 M⊙ (for amorphous carbon grains) with an average dust temperature Tdust = 41 ± 3 K, corresponding to a dust condensation efficiency of 8–12 ${{\ \rm per\ cent}}$. This revised dust mass is up to an order of magnitude lower than some previous estimates, which can be attributed to our different IS dust corrections, lower SPIRE flux densities, and higher dust temperatures than were used in previous studies. The dust within the Crab is predominantly found in dense filaments south of the pulsar, with an average V-band dust extinction of AV = 0.20–0.39 mag, consistent with recent optical dust extinction studies. The modelled synchrotron power-law spectrum is consistent with a radio spectral index αradio = 0.297 ± 0.009 and an infrared spectral index αIR = 0.429 ± 0.021. We have identified a millimetre excess emission in the Crab’s central regions, and argue that it most likely results from two distinct populations of synchrotron emitting particles. We conclude that the Crab’s efficient dust condensation (8–12 ${{\ \rm per\ cent}}$) provides further evidence for a scenario where supernovae can provide substantial contributions to the IS dust budgets in galaxies.