Vermont Senator Bernie Sanders won the Democratic presidential primary in West Virginia on Tuesday, keeping his upstart candidacy alive against frontrunner Hillary Clinton.

Speaking to a rally in a gymnasium in Salem, Oregon, where Democrats vote next week, Sanders conceded that his campaign still faces "an uphill climb" to the left-leaning party's nomination.

"Let me be as clear as I can be: We are in this campaign to win the Democratic nomination," he told a raucous crowd. "And we are going to fight for every last vote in Oregon, Kentucky, California, the Dakotas."

Sanders, a self-described democratic socialist from Vermont, won a week ago in Indiana, while Clinton scored a small victory Saturday in the Pacific island of Guam, a US territory.

With most of the vote counted in West Virginia, Sanders had 51 per cent to 37 per cent for Clinton.

In a state where mining is the backbone of the economy, Clinton's anti-coal comments came back to haunt her. The former US secretary of state and first lady has said that her comment in March about closing coal mines was a "misstatement."

In the race for the Democratic Party's nomination, Clinton has some 1,700 pledged delegates, about 300 ahead of Sanders. The nomination requires a majority, or 2,383 delegates.

Clinton is expected to win an overwhelming share of the 719 so-called superdelegates - mostly elected officials and party insiders - which all but assures her victory at the Democratic convention in July in Philadelphia.

Sanders is hoping to pile up large margins in upcoming states, including Kentucky and Oregon on May 17, to catch up in pledged delegates and make a case for pro-Clinton superdelegates to reconsider.

Sanders on Tuesday night repeated his calls for guaranteed health care, free university education and a massive boost in infrastructure spending.

"If we do well in the coming weeks in Oregon, in California, in New Jersey, in Kentucky and the other states, we still have that road to victory in winning the majority of pledged delegates," he said.

On the Republican side, Manhattan billionaire Donald Trump became the last remaining candidate and presumptive nominee after his last two rivals quit the race after the Indiana primary a week ago.

Trump, who was endorsed by the West Virginia Coal Association, was immediately declared the winner after polls closed late Tuesday in the state. He later scored a similarly unsurprising victory in Nebraska's Republican-only presidential primary.

Sanders won the Nebraska Democratic caucus in March.

Trump's campaign in recent weeks has turned its focus to the general election, and announced Monday that New Jersey Governor Chris Christie - a former presidential rival turned supporter - would lead the campaign's transition team.

The Trump campaign says it has begun shifting toward a general election strategy and "implementing an infrastructure capable of securing a victory, including making key hires, building a finance operation to benefit the Republican Party and unifying the party."

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