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-year, intensive monitoring campaign of the system using 91 observations with the HARPS spectrograph, allowing for detailed modelling of the ∼600 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 m s−1 and Kc = 8.5 ± 2.5 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.