The recent seismicity catalogue of metropolitan France Sismicité Instrumentale de l'Hexagone (SI-Hex) covers the period 1962-2009. It is the outcome of a multipartner project conducted between 2010 and 2013. In this catalogue, moment magnitudes ( M ) are mainly determined from short-period velocimetric records, the same records as those used by the Laboratoire de Détection Géophysique (LDG) for issuing local magnitudes ( M ) since 1962. Two distinct procedures are used, whether M is larger or smaller than 4. For M >4, M is computed by fitting the coda-wave amplitude on the raw records. Station corrections and regional properties of coda-wave attenuation are taken into account in the computations. For M ≤4, M is converted from M through linear regression rules. In the smallest magnitude range M <3.1, special attention is paid to the non-unity slope of the relation between the local magnitudes and M . All M determined during the SI-Hex project is calibrated according to reference M of recent events. As for some small events, no M has been determined; local magnitudes issued by other French networks or LDG duration magnitude ( M ) are first converted into M before applying the conversion rules. This paper shows how the different sources of information and the different magnitude ranges are combined in order to determine an unbiased set of M for the whole 38,027 events of the catalogue. [ABSTRACT FROM AUTHOR]