A series of emails released yesterday appeared to suggest that Smith had been in secret contact with Rupert Murdoch's News Corp about the takeover in 2010 and 2011, and offered assurances of Hunt's support for the deal.
Labour has called for Hunt to resign after claiming that he fell "woefully short" of the standards expected in his role as "quasi-judicial" arbitrator over the takeover.
Hunt, who has insisted that he acted with "scrupulous fairness" during the bid review process, is to give a statement to parliament at 12.30pm.
In a resignation statement, Smith said that his contact with News Corp had been done "without authorisation from the Secretary of State", and admitted that he "went too far" in conversations with News Corp lobbyist Fred Michel.
"While it was part of my role to keep News Corporation informed throughout the BSkyB bid process, the content and extent of my contact was done without authorisation from the Secretary of State," he said in a statement released by Hunt's Department for Culture, Media and Sport.
"I do not recognise all of what Fred Michel said, but nonetheless I appreciate that my activities at times went too far and have, taken together, created the perception that News Corporation had too close a relationship with the department, contrary to the clear requirements set out by Jeremy Hunt and the permanent secretary that this needed to be a fair and scrupulous process.
"Whilst I firmly believe that the process was in fact conducted scrupulously fairly, as a result of my activities it is only right for me to step down as special adviser to Jeremy Hunt."
The culture secretary took over responsibility for ruling on Murdoch's controversial bid to acquire the remaining shares in pay-TV giant Sky in December 2010.
He was handed the power after it was stripped from Liberal Democrat business secretary Vince Cable, who had been secretly recorded saying he had "declared war on Mr Murdoch".
Emails between Smith and News Corp's head of public affairs Fred Michel from June 2010 onwards were released yesterday at the Leveson Inquiry into press ethics and standards, appearing to show that Hunt was broadly supportive of the takeover.
Most shocking was a message dated January 24, 2011, the day before Hunt was due to make a statement to parliament on his intention to refer the takeover bid to the Competition Commission, which contained a message from Michel to News Corp executive James Murdoch claiming that he had managed to get some information in advance on Hunt's statement, adding: "Although absolutely illegal..!"
James Murdoch told the Leveson on Tuesday that the reference had been a "joke".
However, Labour's deputy leader Harriet Harman told the BBC: "I'm afraid there is no explanation as to why it is that, in advance of what Jeremy Hunt said to the House of Commons, the complete strategy and the very words that Jeremy Hunt was going to use, had already been explained to James Murdoch."
Speaking to Sky News outside his home this morning, the culture secretary said: "I have made my position clear. I am going to be making a very, very determined effort to show that I behaved with total integrity, and conducted this process scrupulously fairly."