New satellite analysis from Britain had shown that Flight MH370, with 239 people on board, was last seen in the middle of the Indian Ocean west of Perth, Australia, he said in a statement on Monday.
Flight MH370 disappeared after taking off from Kuala Lumpur on March 8.
The announcement by 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," 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."
Razak appealed to the media to respect the privacy of the families of the passengers and crew, saying the wait for information had been heartbreaking and this latest news harder still.
A text message sent to families by Malaysia Airlines announcing the loss of the plane said: "Malaysia Airlines deeply regrets that we have to assume beyond any reasonable doubt that MH370 has been lost and that none of those on board survived... we must now accept all evidence suggests the plane went down in the Southern Indian Ocean."
RA/BA