Senator McConnell spoke after the electoral college formally confirmed Mr Biden's victory over Mr Trump.

The Democrat won 306 electoral college votes to Mr Trump's 232.

President Trump still refuses to concede, making unsubstantiated claims of widespread fraud.

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.

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.