All 239 people on board were presumed dead, airline officials said.
Analysis of satellite information from British company Inmarsat had shown that the Boeing 777's last position was in the Indian Ocean west of Perth, Australia, Najib said in a statement.
"This is a remote location, far from any possible landing sites," he said. "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."
He added that the families of the passengers and crew had been informed.
"For them, the past few weeks have been heart-breaking. I know this news must be harder still," he said.
Relatives of those on board received the news in a Malaysia Airlines SMS message which said: "We have to assume beyond all reasonable doubt that MH370 has been lost and none of those on board survived."
After the message, there were hysterical scenes at the Beijing hotel where many of the relatives of those on board are staying. More than 150 of the passengers were Chinese.
People wailed, cried and dropped to the floor. One woman shouted out: "It's not possible, it's not possible.
A Reuters reporter on the scene saw at least four people being carried away on stretchers.
China's government immediately demanded that Malaysia share all information and evidence which showed the plane went down in the Indian Ocean.
Najib's comments came as an Australian navy ship was close to finding possible debris from the jetliner after a mounting number of sightings of floating objects that are believed to parts of the plane. The search site is about 2,500 km (1,550 miles) southwest of Perth, in icy sub-Arctic seas that are in one of the most remote parts of the globe.
The objects, described as a "grey or green circular object" and an "orange rectangular object", were spotted on Monday afternoon, said Australian Prime Minister Tony Abbott, adding that three planes were also en route to the area.
Flight MH370 vanished from civilian radar screens less than an hour after taking off from Kuala Lumpur for Beijing on March 8. No confirmed sighting of the plane has been made since and there is no clue what went wrong.
Attention and resources in the search for the plane had shifted from an initial focus north of the Equator to an increasingly narrowed stretch of rough sea in the southern Indian Ocean, thousands of miles from the original flight path.
FLOATING OBJECTS
Earlier on Monday, Xinhua news agency said a Chinese Ilyushin IL-76 aircraft spotted two "relatively big" floating objects and several smaller white ones dispersed over several kilometres.
In a further sign the search was bearing fruit, the U.S. Navy was flying in its high-tech black box detector to the area.
The so-called black boxes - the cockpit voice recorder and flight data recorder - record what happens on board planes in flight. At crash sites, finding the black boxes soon is crucial because the locator beacons they carry fade out after 30 days.
Investigators believe someone on the flight shut off the plane's communications systems. Partial military radar tracking showed it turning west and re-crossing the Malay Peninsula, apparently under the control of a skilled pilot.
That has led them to focus on hijacking or sabotage, but investigators have not ruled out technical problems. Faint electronic "pings" also detected by Inmarsat suggested it flew for another six hours or so, but the initial analysis could do no better than place its final signal on one of two vast arcs, the north and south corridors.
Najib said Inmarsat had been performing further calculations on the data.
"Using a type of analysis never before used in an investigation of this sort, they have been able to shed more light on MH370's flight path," he said.
"Based on their new analysis, Inmarsat and the U.K. Air Accidents Investigation Branch 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."
Asked how Inmarsat experts had made the breakthrough, Chris McLaughlin, senior vice president for external affairs, said: "They tested (the earlier findings) against a number of known flights of other aircraft and came to the conclusion that only the southern route was possible."
The new method "gives the approximate direction of travel, plus or minus about 100 miles, to a track line," he told Britain's Sky News.
"Unfortunately this is a 1990s satellite over the Indian Ocean that is not GPS-equipped. All we believe we can do is to say that we believe it is in this general location, but we cannot give you the final few feet and inches where it landed. It's not that sort of system."