I can think of three ways a river could shoot high into the air:
For my first example, the river begins as a huge lake, high in the mountains, in a natural basin, then flows down through a fissure in the rock, deep underground. It flows underground some distance, then gently bends until pointing directly upwards. With the huge pressure of the lake high above, it jets through a natural nozzle, high into the air. However, there are significant problems with this solution. First, unlike most lakes, a mountain lake does not get much water from runoff, but instead is fed directly by rainfall. The water in the lake would have to be replaced at a rate equal to the rate it is leaving, which means the mountains will have to have substantial rainfall - on the order of monsoon season, all day every day, for even a moderate sized river. Second, the rate of erosion would quickly turn this from a jet pointing upwards to a jet pointing sideways, especially if the water is so highly pressurized that it sprays a mile into the air. Third, the path the water follows would have to be solid rock, capable of withstanding the massive force of the river. In reality, most rock would split, and the river would take multiple paths out, lowering the pressure. This system might be possible if the fissure under the lake opened after the lake was filled and drained the lake drained dry over a period of weeks or even months, but a constantly running stream would be almost impossible. On the up side, this very well might reach a mile into the air, given enough pressure.
The other method would actually be similar to a geyser; water flows down from a natural mountain spring, into an underground passage. As the water moves under the city, it crosses a pocket of magma, which heats the water enough to produce substantial amounts of steam. The resulting increase in pressure would cause the water to jet out of the exit at a much higher rate of flow. However, unlike the previous example, this water would not jet a mile into the air; perhaps only a few hundred feet, at best. The steam, of course, would shoot high into the atmosphere, and could give the impression that the jet of water was much higher than reality. Still, as long as the water maintains a constant volumetric flow, and the magma pocket continues to be heated, you could expect the river to shoot into the air for quite some time. Eventually, of course, the cooling magma and the erosive water would redirect the path of the river, but that may take a century or more.
Finally, my third method uses a bit of engineering and a bit of stage magic. A high natural stream in the mountains is diverted into a huge man-made pipe; the pipe travels down the mountain, and is buried under the city. At the location where the pipe emerges from the ground, the water is funneled into an enormous glass pipe, which reaches high into the air. The pressure forces the water up the pipe, where it then emerges in a geyser. The geyser is only a dozen feet tall, but the water running down the sides of the glass pipe would obscure the pipe itself, giving the impression that the column of water was self-supported, and much higher (and more stable) that would be physically possible. Of all the methods, this is the most likely to remain stable; as long as the mountain stream keeps flowing, and the pipe doesn't break, this system would last for hundreds of years... though don't expect the townspeople to stay in the dark about the glass pipe for very long. It's a tourist trap, not a real magical river.