Dale Heenan, Swindon Borough Council's cabinet member for Highways and Transport, faced a court summons after disputing the amount he owed.
In his resignation letter to council leader David Renard, Mr Heenan admitted he "should not have allowed the issue to proceed so far".
Mr Renard described the matter as "embarrassing" and "difficult".
In a statement, Mr Heenan said he believed he was "entitled to raise questions about the amount being incorrect" and confirmed he has since paid it "in full".
He added the public "is entitled to expect high standards from their elected representatives" and believes "the most honourable course of action" open to him is to "set an example" and resign as a cabinet member.
David Renard agreed that Mr Heenan was right to resign and said: "It's quite clear that the situation isn't one that our councillors should get themselves into.
"We expect all our residents to pay their tax on time and in full and Dale unfortunately fell below the standard that I and most people would expect."
Mr Heenan will remain a councillor for the borough's Covingham and Dorcan ward.
His resignation came about after Private Eye released details gleaned from Freedom of Information requests of councillors across the country who failed to pay council tax.
Only one summons was issued to a member of a Swindon Borough Council, for arrears of Â£981.62.
Although the councillor is not named, in his resignation letter Mr Heenan said he was the councillor who "recent media coverage" concerned.
A Conservative councillor has resigned from a local authority's cabinet over non-payment of a council tax bill.