The system of two transiting Neptune-sized planets around the bright, young M-dwarf AU Mic provides a unique opportunity to test models of planet formation, early evolution, and star–planet interaction. However, the intense magnetic activity of the host star makes measuring the masses of the planets via the radial velocity (RV) method very challenging. We report on a 1-yr, intensive monitoring campaign of the system using 91 observations with the HARPS spectrograph, allowing for detailed modelling of the ∼600 ${\rm m\, s^{-1}}\(peak-to-peak activity-induced RV variations. We used a multidimensional Gaussian Process framework to model these and the planetary signals simultaneously. We detect the latter with semi-amplitudes of Kb = 5.8 ± 2.5 \){\rm m\, s^{-1}}\(and Kc = 8.5 ± 2.5 \){\rm m\, s^{-1}}$, respectively. The resulting mass estimates, Mb = 11.7 ± 5.0 M⊕ and Mc = 22.2 ± 6.7 M⊕, suggest that planet b might be less dense, and planet c considerably denser than previously thought. These results are in tension with the current standard models of core-accretion. They suggest that both planets accreted a H/He envelope that is smaller than expected, and the trend between the two planets’ envelope fractions is the opposite of what is predicted by theory.