He made the announcement on Twitter on Monday afternoon.
He said he feels OK and is not showing symptoms so far, but he has been isolated ever since finding out he may have been exposed.
He warned his followers to stay home and be pragmatic and said he will keep the public updated.
He said that his wife has not been tested yet but is feeling OK.
"This is serious. Now's really the time to think about social distancing, washing your hands," Elba said.
"We live in a divided world right now. We can all feel it," he added. "But now's the time for solidarity, for thinking about each other."
Elba has appeared in such films as "The Wire," "Luther," "American Gangster," "Thor," "Beasts of No Nation" and "Molly's Game."
Last week, Tom Hanks and Rita Wilson said they had also tested positive in Australia.
(The Associated Press contributed to this report.)
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