Bancroft was caught on television cameras appearing to rub a yellow object on the ball, and later said: "I was in the wrong place at the wrong time. I want to be here (in the press conference) because I want to be accountable for my actions."
Smith added: "The leadership group knew about it."
Television footage appeared to show Bancroft, 25, take on object out of his pocket while fielding in the post-lunch session on the third day of the Test at Newlands.
He was spoken to by umpires Nigel Llong and Richard Illingworth during the 43rd over of South Africa's second innings after appearing to have the object in his hand after fielding the ball at cover.
While the umpires were conferring, Bancroft then appeared to place a small yellow object in his underpants.
When the umpires went across to talk to him he reached into a pocket and showed them what seemed to be a different object -- what looked like a soft pouch for sunglasses.
The umpires took no action and did not change the ball.
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There were boos from a capacity crowd at Newlands when the incident was shown on the big screen at the ground.
Both Bancroft and Smith, who took over the Australian captaincy in 2015, appeared before the media after the match and admitted that they had attempted to change the condition of the ball.
"We had a discussion during the break. On myself I saw an opportunity to use some yellow tape and the granules from the rough patches of the wicket to change the condition of the ball," said opening batsman Bancroft who is playing in his eighth Test.
"It didn't work, the umpires didn't change the ball. We have this yellow tape in our kit. The actual sticky stuff itself is very sticky so I felt it could be used to collect some stuff from the side of the pitch."
Smith will now face calls for his resignation after admitting that Bancroft did not act alone.
"The leadership group knew about it. We spoke about it at lunch. I'm not proud of what happened. It's not in the spirit of the game," said the skipper.
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