Denmark’s king announced on Friday night that he was putting a center-right politician in charge of trying to form the next government after Prime Minister Mette Frederiksen failed to build a new coalition.

The news shook Denmark because Ms. Frederiksen has been one of the most dominant Danish political figures in decades and her left-leaning party, the Social Democrats, won the most votes in parliamentary elections in March.

But Denmark’s political landscape has become increasingly fragmented. Smaller parties on the far right and the far left have been gaining steam. Over the past few weeks, Ms. Frederiksen struggled to corral all the opposing forces and couldn’t succeed in hammering together a coalition of leftist and moderate parties.

On Friday night, Denmark’s royal household, led by King Frederik X, issued a statement that said, “The King has requested the Chairman of the Liberal Party, Troels Lund Poulsen, to lead negotiations on the formation of a government that does not involve the participation of the Social Democrats and the Moderates.”

Ms. Frederiksen did not publicly comment on Friday evening.

Most Danes give the prime minister high marks for how she blocked President Trump from acquiring Greenland, a gigantic Arctic island that has been part of the Danish kingdom for more than 300 years.

Yet, when it came to domestic issues like tax policy, immigration and regulations on Denmark’s enormous pig industry, Ms. Frederiksen floundered. In March, her party clocked its worst election performance in a century, winning just 22 percent of the vote. And even though leftist parties won more seats than right-leaning parties, the left-wing coalition came up short of a majority.

A coalition of center-right and right-wing parties will now attempt to form a government. Some have divisive agendas. According to the statement issued by the king, the nationalist Danish People’s Party threw its support to Mr. Poulsen in order to advance “the explicit goal of introducing measures that will lead to Muslim net emigration from Denmark.”

The decisive shift came after Lars Lokke Rasmussen, the head of Denmark’s leading moderate party and, until recently, the country’s foreign minister, walked out of negotiations on Friday. He is now backing Mr. Poulsen, leader of Denmark’s traditional center-right party, which supports free-market economics, lower taxes, farmers and a tougher immigration policy.

In the previous coalition government, Mr. Poulsen served as defense minister. His party won only 10 percent of the vote.

Left-wing parties reacted angrily to the news. Pelle Dragsted, head of the Union List, said on social media, “It is Lars Lokke’s choice,” adding that every party at the negotiation table had been willing to compromise “except Lars Lokke.”

Martin Lidegaard, the head of the centrist Radical Left, said he was “astonished” that Mr. Lokke had aligned himself with the anti-immigration Danish Folk Party and argued that the decision invited “the far right into a central role.”

Danish political analysts predicted that it would not be easy for the right to form a government, either, and that there was still a chance that Ms. Frederiksen could return as the head of some kind of coalition.

“There has still been no movement in the positions that exist,” said Hans Redder, a political analyst at TV2, one of Denmark’s leading news outlets. “They rule each other out across the board, and it is incredibly difficult to see how this will end with a new government. That is where we are now, in what are already the longest government negotiations in Danish history, and I think we can safely add several more weeks to that.”