NEW YORK — Family Dollar Stores Inc., facing a threat from billionaire activist Carl Icahn, adopted a shareholder-rights plan that will limit investors from acquiring more than 10 percent of the company.

All but one of Family Dollar’s directors voted for the plan, which will last one year, according to a statement Monday. Edward Garden, who was added to the board as part of an agreement with shareholder Trian Fund Management in 2011, voted against it.

Family Dollar is adding the so-called poison pill after Icahn disclosed he’d amassed a new 9.4 percent stake and would seek talks with management – a move that sent the shares up as much as 16 percent. He may push for operating changes and ask the company to explore strategic alternatives, as well as potentially seeking board seats, according to a filing.

“Carl Icahn is not someone who’s taken lightly,” Anthony Chukumba, a New York-based analyst at BB&T Corp. who has a hold rating on the stock, said last week. “He has a track record.”

Family Dollar, a chain of budget stores based in North Carolina, has been struggling to compete with rival discounters, drugstores and big-box retailers such as Target and Wal- Mart. To combat slumping sales, the company embarked on a review of its business this year.