PHOENIX — Some 60,000 Arizonans who have been unable to find work are in line for a bit of extra financial help.

Gov. Jan Brewer on Friday signed legislation which alters state law to extend jobless benefits by an extra 13 weeks once the unemployment rate hits 6.5 percent and stays there for three months. Arizona already is there, having first past that figure in December.

And the law adds another seven weeks on top of that once the jobless rate hits and stays at 8 percent for three months.

The state Department of Commerce already is predicting that the figures for this month, to be released in May, will reach that magic number. And Dennis Doby, the agency’s senior director of research administration, is saying they will only go up from there for the foreseeable future.

Brewer had urged lawmakers to make the change once it was determined that any extra cost will be picked up entirely by the federal government. The funds are part of the stimulus package, formally known as the American Recovery and Reinvestment Act, signed into last in February by President Obama.

Arizona provides 26 weeks of unemployment benefits paid for by a tax on state employers. Other federally funded  programs already have extended that another 33 weeks.

Liz Barker, spokeswoman for the Department of Economic Security, said the most immediate beneficiaries will be about 7,000 people already have exhausted their 59 weeks of payments.

“By the end of next week, we will mail out information and an application for extended benefits for those individuals,’’ she said.

Barker said how quickly these people could actually begin seeing new checks depends on how quickly they return those applications and whether anything has changed in their situation that might make them ineligible.

“But they could begin receiving the benefits as early as mid-May,’’ she said.

That group is just the beginning: Barker estimated the total number who collect extended benefits could grow to 50,000 to 60,000 between now and the middle of December when the federal funding dries up.

Both Brewer and Republican legislative leaders have balked at taking other federal stimulus dollars which would expand who is eligible for unemployment compensation. While the initial $150 million cost would be borne by the federal government, Arizona would be required keep the changes — and pick up the financial burden — once the federal dollars go away.

Brewer also signed legislation that spells out all those enrolled in the Arizona Health Care Cost Containment System, the state’s Medicaid program, need to show they remain eligible only once a year.

Lawmakers altered that to every six months for single people without children, a move legislative budget staffers said at the time would save Arizona about $7.5 million a year. But the U.S. Department of Health and Human Services said earlier this year that change, if it stands, makes Arizona ineligible for $1.6 billion in federal aid to help pay health care costs for the poor, including nearly $300 million the state already has spent.