We have developed a new magnitude scale, Mm, based on the measurement of mantle Rayleigh-wave energy in the 50 to 300sec period range, and directly related to the seismic moment through Mm=log10M0 - 20. Measurements are taken on the first passage of Rayleigh waves, recorded on-scale on broadband instruments with adequate dynamical range. This allows estimation of the moment of an event within minutes of the arrival of the Rayleigh wave, and with a standard deviation of ±0.2 magnitude units. In turn, the knowledge of the seismic moment allows computation of an estimate of the high-seas amplitude of a range of expectable tsunami heights. The latter, combined with complementary data from T-wave duration and historical references, have been integrated into an automated procedure of tsunami warning by the Centre Polynesien de Prevention des Tsunamis (CPPT), in Papeete, Tahiti.