It is not a church. Nor can it be counted among the archaeological or natural attractions of the 'Four Corners' region, where the states of Colorado, New Mexico, Arizona and Utah meet in a vista of canyons, jagged escarpments and snowcapped mountain ranges scarcely matched upon the face of this earth.
No, the centre of all the activity is a grey-white building like a warehouse glistening in the brilliant desert light, instantly distinguishable from the dingy homesteads dotting the bare hillsides around it. Outside, a hundred or more cars are clustered beneath signs reading 'By tribal edict: No alcohol or firearms.'
Inside, row upon row of mostly ageing citizens sit transfixed in front of slot machines, drowning their losses in Coke and Sprite. It is, in short, that burgeoning variant on the US craze for gambling, a 'tribal gaming facility' - in plain English an Indian-run casino; the latest attempt by the world's most affluent country to make amends to its original inhabitants.
Nowhere more than here in the south-west do you feel the country's ambivalence about its Indians. Pride in the vanished native civilisations whose traces are to be found in the great national parks near by, of Mesa Verde, Chaco Canyon and Canyon de Chelly, is one thing. The wretched circumstances of their living descendants, for ever a reminder of the indignities, lies and brutalities visited upon them by the settlers in the second half of the 19th century, are quite another.
Few suffered as cruelly as the Ute. 'We shall fall as the leaves from the trees when winter comes,' Ouray, their greatest chief, predicted as he vainly tried in the 1860s to reach a peaceful accommodation with the intruders. 'The lands we have roved for countless generations will be given to the miner and the plowshare, and we will be buried.' And in all but the most literal sense, the Ute have been buried.
Once their territory encompassed the bulk of western Colorado and eastern Utah, an area larger than Britain. But a succession of treaties, each broken after a new gold or silver strike, hemmed them back into two reservations. The one in Colorado is a strip, roughly 15 miles deep and 140 miles long, just north of the border with New Mexico. Even that has been divided. The eastern portion, belonging to the Southern Ute, is administered from a dismal township called Ignacio. Towaoc and the 307,000 acres that surround it are the derisory kingdom of the Mountain Ute.
Time and a guilty conscience have produced some small redress from the conquerors. Indians have become 'Native Americans'. Potentially valuable energy and water rights have been returned; an agreement last week gave the Ute greater control over ancient burial and religious sites on long-lost lands on Colorado's western slope. For the most part, though, life is still the misery of a threadbare existence on the margins of the tourism industry - now the south-west's largest source of income - and of an unemployment rate of up to 50 per cent (depending on whether you believe federal statistics or the tribal leaders).
Of a once nomadic, free-ranging culture, there remain only sour 20th-century symbols; tawdry encampments of mobile homes moored on breeze blocks, and junkyards of automobiles rusting gently into the semi-desert landscape.
And even those resource rights are a mixed blessing. Once, environmentalists and the American Indian made common cause in their defence of a vanishing natural heritage. No longer. A dollars 600m (pounds 380m) project to dam two local rivers, and divert water to irrigate the arid Ute lands, is being fought through the courts by conservationists. Development is held hostage to the Colorado squawfish, said to be in peril because of earlier human tampering with the region's fragile river system. The conquerors continue to believe they know best.
Set against all this, I suppose, the casino is a godsend. Colorado is no different from the 17 other states where, under a 1988 federal law, Indians run gambling operations. Incontestably, they provide jobs and money. The staff at Ute Mountain Casino are mainly native; the punters losing their money are predominantly white - another tiny repayment for the injustices of the past.
The new buildings sprouting around the casino suggest fleeting hope that poverty is not eternal. But the sadness of the place is inescapable. Dutifully, I dropped dollars 5 worth of quarters in a video poker game. As I drove back to the town of Cortez and the white man's West, a forlorn sign informed me I was leaving the land of the Ute: 'We hope you enjoyed the scenery.' Even with the casino, they don't have much else.