MOSCOW—Russian Prime Minister Mikhail Mishustin said he tested positive for the new coronavirus and is temporarily suspending duties in office, becoming the most recent head of government to contract the disease.

Mr. Mishustin, who has served as head of the Russian government since his predecessor, Dmitry Medvedev, handed in his surprise resignation in January, said in a televised videoconference with Russian President Vladimir Putin that the work of the government would continue.

Mr....