Russian President Vladimir Putin has become one of the last world leaders to congratulate US President-elect Joe Biden on his victory.
The Kremlin had said it would wait for official results from November’s vote before recognising the victory.
Democrat Biden won November’s contest with 306 electoral college votes to the Republican incumbent’s 232.
Mr Trump still refuses to concede, making unsubstantiated claims of widespread fraud.
Confirmation by the electoral college was one of the steps required for Mr Biden to take office.
In a blow to Mr Trump, one of his main Republican allies, Senate Majority Leader Mitch McConnell, congratulated Mr Biden.
McConnell, a top member of US President Trump’s Republican Party, congratulated Biden on winning the presidential election last month.
Senator McConnell spoke after the electoral college formally confirmed Mr Biden’s victory over Mr Trump.
Relations with the Senate, currently controlled by the Republicans, will be crucial to Mr Biden’s presidency.
He has arrived in Atlanta, Georgia, to campaign for the Democrats in next month’s Senate run-off elections. Two seats will be decided on 5 January and could determine whether or not his party takes control of the chamber. Democrats already control the House of Representatives.
After Monday’s confirmation of Mr Biden’s victory, three world leaders whose refusal to congratulate the president-elect had been commented widely, did so on Tuesday: Russia’s Vladimir Putin, Brazil’s Jair Bolsonaro and Mexican leader Andrés Manuel López Obrador.
How did McConnell break ranks?
Speaking on the Senate floor, Mr McConnell said he had hoped for a “different result” from the 3 November election but the electoral college had spoken. “So today I want to congratulate President-elect Joe Biden,” he said.
Also congratulating Mr Biden’s running-mate, Kamala Harris, he added: “All Americans can take pride that our nation has a female vice president-elect for the very first time.”
Mr Biden said later he had phoned Mr McConnell to thank him for the congratulations and the two had agreed to “get together sooner than later.”
Senate Democratic Leader Chuck Schumer urged Mr Trump to “end his term with a modicum of grace and dignity.”
“For the sake of our democracy, for the sake of peaceful transition of power, he should stop the shenanigans, stop the misrepresentations and acknowledge that Joe Biden will be our next president,” he said.
Trump’s options running out
It took more than a month, but Mitch McConnell is now acknowledging the obvious – that Joe Biden will be the next US president.
Upon Donald Trump’s departure from the White House, the Senate majority leader will become the senior-most Republican officeholder in the US, so his congratulations to the Democratic president-elect are meaningful.
They will give cover to other members of Congress who may have been holding their tongues lest they draw Donald Trump’s ire. It may allow the presidential transition process to unroll in a more traditional manner – at least where relations between Congress and the incoming administration are concerned.
With Mr McConnell now accepting Joe Biden’s victory, President Trump’s final play – challenging his election loss in Congress on 6 January – also becomes more pointless. Not only will it be blocked by the Democrat-controlled House of Representatives, it seems unlikely to find any traction in the Senate.
Now the waiting for the Trump response begins. Will he shrug off Mr McConnell’s words or add him to the growing list of enemies within his own party that must be confronted and condemned?
The president’s next words and actions could reveal the depth of the political turmoil facing the party in the days ahead.
President Trump does not appear to be changing his stance. Tweeting on Tuesday, he said without offering evidence that “tremendous evidence” of voter fraud was “pouring in”.
When asked if Mr Trump recognised Mr Biden as president-elect, White House press secretary Kayleigh McEnany merely said the president was still involved in ongoing litigation related to the election.
What happens in Georgia?
When Mr Biden takes over the White House on 20 January, he will be looking for the support of both chambers of Congress to carry out his election promises.
Image captionMr Biden addressed a drive-in rally at Pullman Yard in Atlanta
While Democrats control the House, in the Senate Mr McConnell’s party has 52 of the 100 seats. But this could change in January when the two separate run-offs are held in Georgia.
In both races, a Republican incumbent faces a Democratic newcomer: Senator David Perdue, 70, will compete against Jon Ossoff, 33, a documentary filmmaker, while Senator Kelly Loeffler, 50, is up against Rev Raphael Warnock, 51, a senior pastor at an Atlanta Baptist church.
Recent polls suggest Mr Warnock is pulling ahead of Ms Loeffler, while the Perdue-Ossoff race in a dead heat.
If the Democrats win both seats, the Senate will be evenly divided – but it will still be a win for the Democrats. US vice-presidents have the power to cast tie-breaking votes in the Senate, meaning that with Vice-President-elect Harris on board, Democrats would effectively control both chambers.