The federal law required states to set time limits on the receipt of cash assistance and specified that the payments could not exceed five years, which is still the limit in about two-thirds of the states.

In 2010, Arizona cut cash-assistance eligibility to three years, from five. In 2011, it reduced the limit to two years. Then last year it dropped to one, with that limit to begin on July 1.

“The 12-month time limit is rapidly approaching,” said Michael Wisehart, an assistant director at the Arizona Department of Economic Security, which runs the welfare program.

Any cash assistance received since October 2002 counts against the limit, according to state law and letters being sent to welfare recipients. That means even a few months spent on welfare a decade ago can significantly limit future benefits. Arizona officials estimate that 1,200 to 1,600 families — roughly 15 percent of those receiving cash assistance — will be affected. Some may qualify for hardship exemptions.

Domonique Christian, 27, recently received a notice from the state saying that her cash assistance payments of $220 a month might soon be terminated because she had exceeded the new 12-month limit on benefits.

Ms. Christian, a single mother, said she could not afford to lose the aid. She has two young sons and no job. That money, she said, is for diapers, clothing and food for her children, bus fare and other daily expenses.

“The money,” she said, “is really low, but it will get you by. It helps. It’s better than nothing.”

Here, as in several other states, proponents say the strict limits will create a new impetus for welfare recipients to find jobs and will reduce their reliance on public benefits. When President Clinton signed the welfare law in 1996, he said it would replace “a never-ending cycle of welfare” with “‘the dignity, the power and the ethic of work.”