(CBS) This first year, he has run up a record as a pro-business, economic conservative with liberal views on social issues: guns, gay rights, the environment. He managed to reform the state's costly Workman's Compensation law. But on the issue that got him elected, California's huge budget deficit, the best he could do was refinance $15 billion in state debt. He was unable to get significant spending cuts through the legislature and was dead set against raising taxes.

He has taken some heat over the issue of balancing the budget. "That you, in the course of trying to pay the bills, you got deeper in debt," says Safer.

"Did we solve the problem in one year? No," says Schwarzenegger. "They have ruined the state for five years. They kept spending, spending and spending. We're talking here about addicts. And the people sent me to Sacramento to be the outside intervention."

But the honeymoon with voters and the state legislature came to a crashing halt earlier this spring. Democrats and public employee unions went on the attack over the governor's proposed belt-tightening. They dogged his steps with protests, and flooded the state with critical TV spots.

Schwarzenegger's approval rankings sank 20 points in three months, forcing him on the defensive.

Showing 60 Minutes around the governor's domain in the Capitol, he takes a kind of boyish pride in his Wall of Fame, great moments in the Schwarzenegger administration. In his office, he keeps his favorite movie weapon: the sword from "Conan the Barbarian," the film that made him a star.

He wheels and deals in his smoking tent, set up in an open-air courtyard near his office. It's a place to schmooze and cajole, light up the biggest cigar in the world and, for good measure, outrage health advocates.

His wife, Maria Shriver, has been a high-profile first lady, active in women's causes. But the governor is quick to point out their political differences.

When he showed Safer his wife's office, he said, "I'll show you Maria's office quickly, which is back here. She, of course, has to be further away from my office because she's a Democrat."

She's not just any Democrat, but a Kennedy.

That made it all the more unusual this summer when she went where no Kennedy had ventured before: The Republican National Convention, where she watched her husband wow delegates with a tried-and-true Schwarzenegger-ism: "And to those critics who are so pessimistic about our economy, I say, don't be economic girlie men!"

The governor parts ways with the president, and with his own California Republican party, on a number of issues — none more controversial than embryonic stem cells.

"I'm a religious Republican, too. I'm a Catholic. I go to church every Sunday," says Schwarzenegger. "But it doesn't mean that I'm against progress. We have to move forward with the stem cell research. This is the future of solving some of the serious problems that are existing in a medical area. And, why not?"

Normally, even a first-term California governor with Schwarzenegger's poll numbers and star power would be talked about as a potential future presidential candidate. But being foreign-born, he can't run. There are attempts to change that.

Would he like to be able to? Would he like to see an amendment to the Constitution?

"Yes. Absolutely," says Schwarzenegger. "I think, you know, because why not? Like with my way of thinking, you always shoot for the top. But it's not something that I am preoccupied with. I am not thinking one single minute about that. Because there's so many things I have to do in California, and my promise was to straighten out the mess in California."

60 Minutes concluded its visit with The Governator by showing him the last few seconds of their first encounter 27 years ago, back when Schwarzenegger was a 30-year-old on a very fast track.

Back then, Safer reported: "Arnold Schwarzenegger is a contented soul. He's happy as a clam, strong as a horse. And if you find bodybuilding a funny thing for a grown-up, intelligent man to be doing, so does Arnold. And he's laughing with you — all the way to the bank."

What does Schwarzenegger think of that today?

"That's funny. It's amazing to see that. So you have to get me a tape of that," says Schwarzenegger. "And I'm still laughing. Not all the way to the bank. My wife is screaming, 'Now you make no money because ...' But at least I have a good time. And it's a challenging job. And I am having the greatest time in the world doing this."

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