In a video posted to Twitter, the 47-year-old actor said he received his test results on Monday, but he feels "OK" and is showing no symptoms.
He added that he has been isolated since he found out that he has been exposed to someone infected with the virus.
Elba encouraged people to “stay home and be pragmatic”.
More celebrities infected with Covid-19
Also on Monday, Kristofer Hivju, a Norwegian actor who starred in ‘Game of Thrones’, announced that he had tested positive for the illness.
“I urge all of you to be extremely careful; wash your hands, keep 1,5 meters distance from others, go in quarantine; just do everything you can to stop the virus from spreading,” he wrote on Instagram.
On Sunday, a former James Bond actress, Olga Kurylenko, said she has contracted the virus, and asked her fans to take the pandemic seriously.
"Fever and fatigue are my main symptoms. Take care of yourself and do take this seriously!" she wrote in an Instagram post.
Last week, Tom Hanks announced that he and his wife had tested positive for coronavirus.
The World Health Organization (WHO) declared the virus a global pandemic.