The vote came during a special called meeting April 7 following Goodwin's April 1 written resignation.
Council Member Andrea Willott said she wanted to clarify that Goodwin did not identify a term from which he resigned, and the April 7 action is only effective through the end of April, as Goodwin is still running unopposed for the upcoming election.
Council Member Jon Cobb also pointed out that even though he resigned from City Council verbally on March 28 following a heated exchange between Cobb and Goodwin, Cobb is still attending council meetings so the body can continue to operate under a quorum.
City Manager Clint Garza had explained earlier that council can only accept written, not verbal resignations.
Following the unanimous vote to accept the resignation, Acting Mayor Kara King said she welcomed the opportunity for Goodwin to come before council some time in the future so the body may thank him publicly for his service.
Later during the April 7 meeting, council also discussed appointing someone to replace Goodwin.
Three people submitted applications for the position, and following discussion of the item, applicant Kevin Hight, who also serves on the Bee Cave Planning and Zoning Commission, was appointed by council through a unanimous vote to fill Goodwin's seat.