"We'll never get their attention," sighs one of the harassed teachers accompanying the group, "until we find out if he is here or not. All they've been talking about on the way here is whether or not they will get to see 'Arnie'. It's pathetic."

Pathetic or not, just one year after taking office, California's celebrity governor is now a big attraction in Sacramento, where tourism figures have increased for the first time in years and visitors to the Capitol building have soared. To the teacher's evident relief, she discovers that the governor is out of town. The girls content themselves with being photographed next to the highway-patrol trooper guarding the double doors to the governor's office. Schwarzenegger caused his name to be added in prominent gold letters above the word "Governor", engraved in marble over the doors, for no other reason, according to Margita Thompson, his press secretary, than to improve the photo opportunity.

Behind the doors, a lobby acts as a waiting room and connects to "the Horseshoe" — a U-shaped arrangement of offices facing onto a courtyard, wherein may be found a brown canvas tent pitched on improbably green Astroturf. This is Arnie's "smoking tent", a refuge from the rigorously non-smoking building, where he can enjoy a cigar, despite complaints that fumes seep into the windows of the offices above.

"Actually, he does a lot of work, signs a lot of bills, in there," says Patricia Clarey, his amiable chief of staff, whose own office has a fine view of the tent. On one wall is a photographic patchwork of movie stars who have come visiting since Schwarzenegger became governor. "This is not like a normal government office," Clarey says. "I can be sitting here working and get a call to say that Danny DeVito or Jamie Lee Curtis has just walked in. His Hollywood friends want to come up and see what he is doing. He makes the job fun for himself and fun for us."

"Fun" is the word most bandied around in the Governator's administration. But perhaps he gets the greatest fun from confounding the cynics who, last year, were asking how this immigrant body builder and movie star, with a rackety past and without a day's experience in government, could take control of the world's sixth largest economy and a state facing serious financial problems. How wrong they were. Governor Schwarzenegger has won much better reviews for his first year in office than he ever got as a movie star. "He radiates hope, confidence and optimism, all the qualities people look for in a leader," says Jonathan Wilcox, a speech writer for the former governor Pete Wilson. "Californians had begun to worry that things would never get better. Now they believe things are looking up."

Schwarzenegger will celebrate the first anniversary of his inauguration on Wednesday, knowing he is one of the most popular politicians in the US. It was telling that in the final days of George Bush's ultimately successful election campaign, Schwarzenegger was wheeled out to provide celebrity ballast: although he and Bush have little in common politically, he was seen by Republican strategists as a valuable counterweight to John Kerry's endorsement by Bruce Springsteen. He campaigned for Bush only in the key battleground state of Ohio. Certainly in California, the combination of celebrity, personal charm, business acumen and intelligence has proved unbeatable.

In ballots held at the same time as the presidential elections, California voters backed their governor over three key issues. They approved $3 billion in borrowing to fund stem-cell research (deeply unpopular with Bush and the right), opposed the relaxation of hard-line criminal justice policies, and supported tighter controls of the gaming industry. They may have rejected his call to throw out the controversial proposition 63, a measure to tax millionaires by an extra 1% and invest the proceeds in mental-health care, but he still has a powerful mandate.

No wonder there is more and more talk within the Republican party of trying to push through an amendment to the constitution that would allow him to stand as a presidential candidate in 2008, when Bush will stand down under the rules that allow presidents to serve only two terms. Schwarzenegger brushes aside such talk, claiming he has enough to do "fixing California", but nobody underestimates the giddy heights of his ambition, and a number of political analysts to whom I spoke made the same point. "Where do you go after being governor of California? Obviously, the White House."

Covering Schwarzenegger on the campaign trail is an experience like no other. Where else would you find teenagers in miniskirts queuing to listen to a speech on workers' compensation? Where else would you see people shaking his hand, then running to the back of the crowd to get in line to shake it again? What other governor flies from event to event in his own Gulfstream V jet?

Last month, Schwarzenegger launched a series of "Ask Arnold" meetings across the state. They were billed as "town hall" meetings, as if they were open to all; in fact they were invitation-only affairs and were designed to garner support from the business community to defeat two proposals before the legislature about gaming that Schwarzenegger deemed undesirable. The first meeting was held at the Hyatt Regency hotel in Irvine, a prosperous town in Orange County. All the men turned up in suits and ties, and the women arrived in power suits or dolled up as if for a cocktail party — shimmering gowns, many much too short, plunging necklines, stilettos and full make-up. Nobody appeared to know what the meeting was about. "Who cares what it's about?" asked a large blonde in vermilion crepe de chine and matching lips. "Listen, honey, you get an invitation from Arnold, you go. Right?"

As Schwarzenegger emerged from behind a screen, the audience got to its feet to applaud. Grinning broadly, tanned, handsome, he wore a white open-necked shirt and a windcheater embroidered with the numbers of the propositions he wanted to defeat, crossed through like road signs. "Tank you," he began, "tank you for de vunderful velcome." His accent is so preposterous that one wonders if, off stage, he talks like other Californians. I am assured by his aides that this is not the case.

SCHWARZENEGGER FOR PRESIDENT?