Sky said last week that any team members, including riders, staff and management, who refused to sign a document saying they have never doped would be sacked.
And the American, who took third place in the 1998 Tour de France, has disclosed having taken a banned substance as a professional rider.
Team Principal Dave Brailsford said: “Bobby has shown courage in admitting to the errors he made long before his time with Team Sky. We understand that this is a difficult step for him and we’ve done our best to support him.
Wiggins doubts Armstrong will admit to cheating
"It’s important to emphasise that there have been no doubts about his work with us or his approach as a coach. He has done a good job and been a good colleague during his two years with us. Bobby has our best wishes for the future.
“We’ve made clear our commitment to being a clean team and been open about the steps we’re taking. Although it’s never easy to part, we believe this is the right thing to do.”
Sky, whose team includes Tour de France winner and multi-Olympic gold medallist Bradley Wiggins, took the step to erase any suspicions about their team in the wake of the Lance Armstrong scandal.
The American was banned for life and stripped of his seven Tour de France titles after deciding not to answer charges that his success was fuelled by performance-enhancing drugs.
Armstrong has denied any wrongdoing but USADA said the case against him, which included sworn testimony from 26 people, was undeniable proof he had cheated.
Contador: Armstrong being "humiliated and lynched"
Tour de France
UCI and USADA escalate war of words over Armstrong