Footage shot by Louis Tomlinson, 22, shows him passing the cigarette to Zayn Malik, 21, while travelling in a car through Peru.
"So here we are, leaving Peru. Joint lit. Happy days," Louis can be heard saying in the video, posted on the Mail Online.
Louis Tomlinson and Zayn Malik of One Direction are heard in the video joking about smoking a "joint"
He repeatedly jokes about the police, or "po po", and as an officer drives by, says: "He's having a look, he's thinking I'm sure I can smell an illegal substance in there. And he's hit the nail on the head."
The leaked video could damage plans to draw on the band’s success in America. If it is proven that the pair have taken illegal substances, they could be banned from the US, where they are due to tour this summer. Under Peruvian law, using small amounts of cannabis for personal use is not punishable.
At another point in the video, Louis focuses the camera on his bandmate, saying: "That's just Zayn warming up before the show. Zayn takes his job very seriously. He makes sure he goes through a two-hour intense warm up regime before every show, just to get himself in the zone.
He continues, referencing the slang name for cannabis: "One very, very important factor to Zayn's warm up, of course, is Mary Jane herself. In fact, I'm presenting it to him now for some fantastic singing."
Louis later ponders: "I'm just wondering now, we're sitting here in Peru. Will this come back to me? Who knows? Maybe. Maybe not."
Liam Payne, Harry Styles and Niall Horan, all 20, were not in the SUV with their bandmates when the video was shot.
Bandmates Harry Styles, Liam Payne and Niall Horan were not in the vehicle when the incident took place
A spokesperson for Columbia Records, who represent the band in the UK, said: "Our only comment at the moment is the whole situation is in the hands of our lawyers."
One Direction have just returned from South America and are due to play at Sunderland's Stadium of Light on Wednesday night before touring Britain and Europe. In August they are scheduled to tour Canada and North America.