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In a transcript released by the Kremlin, Mishustin said he will continue to govern remotely, by phone and by video.
"It just became known that the tests that I passed for coronavirus gave a positive result," he said. "In this regard, in accordance with the requirements of the Rospotrebnadzor, I must observe self-isolation, follow the doctors' orders. This must be done to protect my colleagues.
"The government will continue to operate normally," Mishustin continued. "I plan to be in active contact with colleagues by telephone and video, with you, Vladimir Vladimirovich, on all basic issues. And as an interim, I propose the candidacy of Andrei Removich Belousov."
Putin offered his condolences and saluted Mishustin for his loyal service to the government.
"What is happening to you now can happen to everyone. I always talk about it," he began. "You are a very active person, I want to thank you for the work that has been done so far. You and the members of the Government, colleagues from the Presidential Administration, are certainly in a zone of particular risk, because no matter how you limit yourself in contacts, you can’t do without direct communication with people when making decisions. And here, of course, there is nothing surprising."
Putin assured the prime minister that any major decisions concerning the coronavirus and the economy would not be made without his input.
"I want to tell you, without your opinion, without your participation, these final decisions will not be taken," he added. "I want to wish you recovery as soon as possible. Get well soon."
Mishustin then asked all Russian citizens to take the viral threat seriously and encouraged them to stay home.
"May holidays are ahead, and I ask all of you to stay at home and comply with all necessary restrictions," he said. "I'm sure that together we will be able to defeat this infection and return to normal life. The government is taking and will take all necessary steps for this."
Putin accepted Mishustin's nomination of Andrei Removich Belousov for the position of Acting Chairman of the Government of the Russian Federation, before wishing him good luck and a speedy recovery.
As of Thursday morning, Russia had the eighth largest number of COVID-19 cases in the world with over 106,000 but has only reported around 1,000 deaths.