Najib Razak said this was the conclusion of fresh analysis of satellite data tracking the flight. Malaysia Airlines had told the families of the 239?people on board, he said.
Earlier the BBC saw a text message sent to families by the airline saying it had to be assumed ?beyond reasonable doubt? that the plane was lost and there were no survivors.
Flight MH370 disappeared after taking off on 8 March from Kuala Lumpur.
The announcement by PM Najib Razak, at a late-night news conference, came on the fifth day of an international search effort in the southern Indian Ocean.
Based on new analysis, the UK?s Air Accidents Investigation Branch and Inmarsat, the UK company that provided satellite data, ?have concluded that MH370 flew along the southern corridor, and that its last position was in the
middle of the Indian Ocean, west of Perth,? Mr Razak said.
?This is a remote location, far from any possible landing sites. It is therefore with deep sadness and regret that I must inform you that, according to this new data, flight MH370 ended in the southern Indian Ocean.?