BEFORE WE CLIMB INTO THE BOATS, Jim Norton tells a story that he calls "Four and a Half Billion Years in Twenty Minutes." This basically consists of forming the sand on the bank of the Yangtze River into the outlines of India and China, then smashing the two countries together with his feet. The result is a miniature version of the 13,000-foot-high mash-up known as the Tibetan Plateau. Patty-caking his way around the edges, Jim raises the Himalaya, Pamir, and Kunlun ranges, closing off three sides of this, the roof of the world.

What's a drop of water to do but flow east? Four of the planet's greatest rivers—the Yangtze, Salween, Mekong, and Irrawaddy—exit Tibet the only way they can, dropping off into southwestern China's verdant Yunnan province, or, in the Irrawaddy's case, Myanmar. Add hundreds of millions of years of tectonic grinding—Jim does this with his hands—and the result is what he calls Yunnan's "train wreck of geography," a bent and fractured landscape where all four rivers twist and reverse themselves, each divided from the next by 20,000-foot ranges of sandstone and limestone, flowing through a choke point just 56 miles wide. Tibetans call this spot the Mother of Waters: One in 12 people lives downstream from this narrow passage. Not one in 12 people in China, or one in 12 in Asia—one in 12 people on earth.

Clean-shaven and with the rounded shoulders of a longtime river guide, Jim, 34, molds and lectures, hurrying through Tibetan geology, Sichuan morphology, Burmese hydrology, and the human and biological diversity that has arisen over millennia in this crenellated, inaccessible region. Roughly the size of Germany, Yunnan is a hothouse of 20 endangered animals and 6,000 rare plants, containing everything from snow leopards and golden monkeys to 70 percent of the herbs in the Tibetan apothecary. This bounty is nominally protected by a series of reserves and proposed parks the size of West Virginia, a joint initiative of the Chinese government and the Nature Conservancy called the Yunnan Great Rivers Project. But conservation is a delicate business here: Wild rivers produce as much hydropower as biodiversity, and China is starving for cheap electricity. Jabbing one long finger into his ankle-high model, Jim marks where the Chinese government plans to build a massive hydroelectric dam. Then he does it again: another cut with his finger, another dam. And again. And again. And again.

"Not dams," Jim clarifies, his fingers trailing in the dirt, "but entire suites of dams, along with your standard sweep of insults, like road building and deforestation." Twelve or more dams are planned for this river; 13 more, including two that have already been built, will cascade down the headwaters of the Mekong, with still more on the Salween and Irrawaddy. That's just the beginning: Some government officials estimate up to 100 more in coming decades.

Jim wraps up his lesson in the advertised 20 minutes, and then kicks his sand creation into the river. The granules are starting a considerable journey, for this is the third-longest river in the world, and known by many names. Up in Tibet they call it Tongtian Ho, "the Way to Heaven." Here in Yunnan, it tumbles past us in a low, wintry torrent called the Jinsha Jiang, or "Golden Sand River." Seven hundred miles downstream, where it passes through the floodgates of the still-rising Three Gorges Dam and into the superheated valleys of China's lowlands, it is the Chang Jiang—the "Long" or "Eternal" river. Only after it passes Shanghai, 3,900 river miles below its source, does this water become the Yangtze, "Son of the Sea."

I put my toe in the verdigris water for the first time. Cold, for a sunny day in the tropics, but the river is snowmelt and we are at 8,000 feet. Just upstream is Tiger Leaping Gorge, a black thundercloud of Himalayan rock seemingly split in half by God himself. Every schoolchild in China knows the beauty of this impassable river canyon, its rock walls rising from frothing waters straight up into the clouds. Even here, on a gravel beach a mile downstream, the violence of the water's passage rattles the air. Although the gorge is a World Heritage site, protected by UNESCO rules, the first dam will go right at its top.

In 1986, a group of Chinese adventurers decided to preempt any foreigners and make a first descent of Tiger Leaping Gorge. Two teams of paddlers launched into the Class VI+ rapids; nine men died. A small museum upstream celebrates their heroism and shows their demented craft, a capsule made from two airliner-style rafts lashed together and covered with spare tires. What the Chinese learned from this highly publicized triumph is that whitewater rafting can be fatal.

That's why we're putting in below the gorge. Our group of 21 Chinese and American clients and guides push and strain at five rafts, launching into the flicking current. Below us, the river is swallowed by the 300-mile Great Bend, a passage through rugged mountains, seldom-visited villages, and 12,000-foot walls that is China's answer to the Grand Canyon. We'll run 128 miles, drop a thousand vertical feet, and float Class IV rapids that may soon be submerged.

Organized by Mountain Travel Sobek, ours is by no means the first commercial float down this stretch of the Yangtze—whitewater explorer Richard Bangs led the initial commercial trip down the Great Bend in 1987—but it's part of the first explicit attempt to create something enduring from this vanishing resource, the fruit of years of effort by Jim and his fellow guides to demonstrate a sustainable alternative to some of the dams. Of course, they'll have to paddle hard to outrun this wave of concrete: Although the dams are currently under review in Beijing, Yunnan's Communist Party secretary, Bai Enpei, insists that almost all will be built.

We slide down into the first rapids and are enfolded by rising walls. Time, and the river, are running.