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The last measurement of the diffuse emission spectrum of the Milky Way in the megaelectronvolt (MeV) photon energy range was performed by CGRO/COMPTEL more than 20 years ago. We report a new analysis with the spectrometer SPI aboard INTEGRAL in the band 0.5-8.0 MeV, finally superseding the signal-to-noise ratio of the historic observations. This is possible thanks to an elaborate instrumental background model and careful considerations of the selected data, which are strongly affected by solar activity. We base our analysis on energy-dependent spatial template fitting in a region of $Δl \times Δb = 95^\circ \times 95^\circ$ around the Galactic centre. Our flux estimates are consistent with COMPTEL measurements and show no `MeV bump'. The spectrum follows a power-law shape with index $-1.39 \pm 0.09_{\rm stat} \pm 0.10_{\rm syst}$ and an integrated flux of $(5.7 \pm 0.8_{\rm stat} \pm 1.7_{\rm syst}) \times 10^{-8}\,\mathrm{erg\,cm^{-2}\,s^{-1}}$ between 0.5 and 8.0 MeV. We find that cosmic-ray electrons and propagation models consistent with the latest Fermi/LAT, Voyager 1, and AMS-02 data are broadly in agreement with the inferred inverse Compton spectral shape. However, a mismatch of a factor of 2-3 in normalisation with respect to baseline expectations may point to enhanced target photon densities and/or electron source spectra in the inner Galaxy, slightly modified diffusion properties, or the presence of an unresolved population of MeV $γ$-ray sources.