The governor, who has been a leading voice among Republicans in warning of the dangers of the virus, tested positive while being screened to greet President Trump in Cleveland.

Gov. Mike DeWine left his home in rural Ohio at about 6 a.m. on Thursday, setting out early to meet President Trump when he arrived for a visit in Cleveland.

During the three-hour drive up Interstate 71, Mr. DeWine prepared to speak to the president about an issue that had been on his mind: coronavirus testing. But before any discussions, the White House required him to get his own rapid-result test. He thought little of it until the results came back with unwelcome news.

Mr. DeWine, 73, had tested positive, becoming the second governor known to do so as infections spread across the country.

Though other Republicans and people close to Mr. Trump have tested positive, Mr. DeWine has struck a distinctly different tone than some in his party, taking an early and aggressive response to the virus and regularly wearing a mask. Critics within his own party quickly seized on the news as a sign that the strict measures he put in place, including a mask mandate, had not worked.

Mr. DeWine, who was waiting for the results of a second test on Thursday, said he did not know where he may have contracted the virus and had no symptoms other than a headache.

“The lesson that should come from this is, we’re all human,” he said, rejecting assertions that masks were not effective. “This virus is everywhere.”

Mr. DeWine was tested as part of a standard protocol in preparation for Mr. Trump’s visit to Ohio on Thursday. He did not meet with the president and left to get a secondary test.

“A very good friend of mine just tested positive,” Mr. Trump said in front of Marine One in Cleveland, before departing to visit a Whirlpool factory in Clyde, Ohio. He praised Mr. DeWine for doing a “fantastic job.”

“We want to wish him the best,” Mr. Trump said. “He’ll be fine.”

Mr. DeWine’s positive test result reflects the growing threat the coronavirus poses in states like Ohio, which is now averaging about 1,200 cases a day, more than the state’s previous peak in April. It is also the latest reminder of a point Mr. DeWine had made this week: The virus can spread quietly and widely, reaching even those who take the threat most seriously.

This afternoon, Governor DeWine talked with local journalists about testing positive for #COVID19. Watch the entire Q&A here: https://t.co/fw0YouHYwP pic.twitter.com/wl2U8s9jbz

— Governor Mike DeWine (@GovMikeDeWine) August 6, 2020

Several people have tested positive as part of regular screenings meant to protect the president. The easy access to tests and immediate results stand in contrast to the experience of many Americans, who have had to wait hours to get tested for the virus and continue to face turnaround times that stretch for days and even weeks, far longer than the 24 to 48 hours that experts recommend to effectively quarantine and contact trace to stop the spread of the virus.

Representative Louie Gohmert, a Texas Republican who has frequently refused to wear a face covering in the Capitol, said he tested positive for the coronavirus at the White House last week, before a planned trip with Mr. Trump on Air Force One. Campaign staff members who attended the president’s rally in Tulsa, Okla., in June also tested positive.

Separately, Gov. Kevin Stitt of Oklahoma, a Republican, became the first governor known to test positive for the virus last month, a diagnosis he said was unrelated to his attendance at the president’s rally.

Among those Republicans who tested positive, Mr. DeWine stands out as the most ardent disciple of public health measures, taking a different tack than Mr. Trump, who initially denied that the virus was a threat and for months resisted wearing a mask in public. (The president wore a mask when visiting the Whirlpool factory on Thursday.)

Mr. DeWine was the first governor to shut down schools and issued an early stay-at-home order in March. He wakes up each morning to a fresh PowerPoint presentation from his staff that he reads on his iPad before 8 a.m. He issued a statewide mask order last month and often issues dire pleas to Ohioans to take the virus seriously.

“Don’t we all want to be around to meet our future children, our future grandchildren?” he said during a televised state address last month. “To attend their baptism, to watch our kids and grandkids graduate from school?”

A mild-mannered career politician, Mr. DeWine spent decades in public office — as a county prosecutor, state senator, congressman, lieutenant governor, U.S. senator and state attorney general — largely out of the national spotlight, until the pandemic turned him into something of a social media sensation.

His daily 2 p.m. press briefings, where he took on a professorial air, speaking alongside graphs and charts while wearing round glasses and colorful ties representing Ohio universities, spawned a fan club. The briefings, known as “Wine With DeWine,” inspired T-shirts and wine glasses with the motto “It’s 2 o’clock somewhere.”

But his approach also drew uproar from protesters who gathered outside the State Capitol and from members of his own party. Amid the stay-at-home order and business closures, Republicans accused his administration of “micromanaging” residents and pumping up coronavirus statistics to scare Ohioans.

On Thursday, a Republican state representative, Nino Vitale, used the moment to question the governor’s policies. “I thought masks worked?” he posted on Facebook, alongside a photo of Mr. DeWine wearing a face covering.

By Thursday afternoon, Mr. DeWine said he had received several “not so nice” text messages suggesting that mask wearing did not matter.

“Look, we know it does,” he said, speaking from the porch of his home in Cedarville, Ohio. “If people take that lesson from the fact that I apparently have it, that would be the wrong lesson.”

In addition to being in a higher-risk age group, Mr. DeWine has had asthma since he was a teenager. As he and his wife, Fran, waited for further test results, he said he planned to work from home and quarantine for 14 days. Ohio’s lieutenant governor, Jon Husted, was also tested for the virus on Thursday and received a negative result.“So far, my work is not going to be impacted,” Mr. DeWine said. “We’ll see. I don’t take anything for granted. This virus is a nasty thing.”