The ordinance allows stores to provide recycled paper bags at the customer's request for a fee of no less than 5 cents. It requires stores to make reusable bags available for purchase after Jan. 1, 2012.
The ban applies to grocery stores, pharmacies and convenience stores of at least 10,000 square feet.
Plastic bags given by pharmacists for prescription medications and bags without handles that are used to protect purchased items from damaging or contaminating other items are exempted from the ban.
Marin County Deputy County Counsel David L. Zaltsman said more than a dozen people spoke in favor of the ban at Tuesday's meeting, and two people spoke against it.
Stephen Joseph, an attorney for the Save the Plastic Bag Coalition, said he plans to file a suit challenging the ordinance in the next two days in Marin County Superior Court.
The group claims paper bags have more negative environmental impacts than plastic bags.
The Board of Supervisors delayed voting on the ban on Jan. 4 after it received hundreds of pages of documents from the coalition.
The coalition said the county failed to comply with the California Environmental Quality Act when it passed the ordinance, and should have reviewed the ban's environmental impacts first.
Zaltsman said the ordinance is exempted from complying with the act because it is a regulatory measure designed to protect natural resources and the environment.
He told the board that banning plastic bags will have a positive environmental impact as long as customers don't simply switch to single-use paper bags.
Zaltsman said he believes even a nickel charge for a paper bag will convince customers to switch to reusable bags.
First-time violators of the ordinance will receive a written warning. Fines ranging between $135 and $440 will be imposed for additional violations.
The town of Fairfax also has banned single-use plastic carry out bags.