The deep structure of the Moon is a missing piece to understand the formation and evolution of the Earth–Moon system. Despite the great amount of information brought by the Apollo passive seismic experiment (ALSEP), the lunar structure below deep moonquakes, which occur around 900 km depth, remains largely unknown. We construct a reference Moon model which incorporates physical constraints, and fits both geodesic (lunar mass and polar moment of inertia, and Love numbers) and seismological (body wave arrivals measured by Apollo network) data. In this model, the core radius is constrained by the detection of S waves reflected from the core. In a first step, for each core radius, a radial model of the lunar interior, including P and S wave velocities and density, is inverted from seismic and geodesic data. In a second step, the core radius is determined from the detection of shear waves reflected on the lunar core by waveform stacking of deep moonquake Apollo records. This detection has been made possible by careful data selection and processing, including a correction of the gain of horizontal sensors based on the principle of energy equipartition inside the coda of lunar seismic records, and a precise alignment of SH waveforms by a non-linear inversion method. The Very Preliminary REference MOON model (VPREMOON) obtained here has a core radius of 380 ± 40 km and an average core mass density of 5200 ± 1000 kg/m3. The large error bars on these estimates are due to the poorly constrained S-wave velocity profile at the base of the mantle and to mislocation errors of deep moonquakes. The detection of horizontally polarized S waves reflected from the core and the absence of detection of vertically polarized S waves favour a liquid state in the outermost part of the core. All these results are consistent, within their error bars, with previous estimates based on lunar rotation dissipation (Williams et al., 2001) and on lunar induced magnetic moment (Hood et al., 1999).