When building a stone arch bridge, the structure needs to be placed on solid ground if possible. Usually, if you dig down far enough, good firm ground can be found. Every once in a while, though, solid, non-compressible ground cannot be readily found. Fortunately, there are several viable options for building a stone bridge in this type of scenario.
Flaring the Base
One very simple method of reducing the amount of settling a stone arch bridge may experience when built on soft ground is to flare the foundations out underground. Basically, the foundation is built significantly wider than the upper portions of the bridge require. Then, as the following courses of masonry are laid, each course is made a little narrower than the one below until the foundation has been tapered to the correct size for the actual bridge. Doing this greatly increases the area over which the weight of the bridge is distributed, and can significantly reduce chances of settling.

A related alternative available, thanks to modern construction materials, is to pour a wide, reinforced concrete slab below the ground upon which the bridge rests. Two large slabs can be poured, one for each abutment, but a more foolproof alternative is to pour a single slab large enough for both abutments to rest on. Such a design almost guarantees that the two abutments will not settle interdependently. This is advantageous, as independent settling of the abutments is the most harmful type of settling for a stone arch bridge.
Firming up the Ground with Piles
A rather unusual way of increasing the load-carrying capacity of the ground is to drive in numerous wooden piles at the bottom of the excavation for the bridge abutments. This method was used extensively when building historic stone arch bridges, and can be successful. Basically, the bridge rests largely on the wooden piles, which go deep into the ground. There is a large amount of friction between the wooden piles and the ground, which helps prevent settling. The wooden piles last a long time underground, as no air can access them; hence they are much less vulnerable to decay. Often, a wooden platform was built atop the piles and the bridge founded on it. Strange as it may seem, this method of stone arch bridge building does work, as numerous examples built this way worldwide can testify. The main threat with this design is scour. A more modern alternative to this method of founding the bridge on wooden piles is to use steel or, better yet, reinforced concrete.
Creating Hard Ground
Some soils can be made quite hard with a little packing. Clay is particularly easy to pack into a solid mass. Another possibility is to pack gravel into the soil below. When done with clay, the end result is rather similar to concrete in some ways. A historic method of taking this one step further is to pour a large amount of gravel into the excavation where the bridge foundations are to be located, and build the bridge atop the gravel. Still, this can be somewhat limited; if the ground is very soft the gravel can sink, bridge and all.
Limitations to Building on Soft Ground
One or a combination of the above-listed methods can be used with success to prevent settling. However, scour is a very real threat for bridges built on soft ground. It is worth noting that scour can be prevented (see 3 Ways to Prevent Scour) and that some of the methods mentioned above, with some adaptions, can actually be used as scour barriers in their own right. Take, for instance, founding the bridge on reinforced concrete piles. This is rather similar to how scour-prone bridges are underpinned, though, of course, it is a little more complicated to place piles under an existing bridge than it is to build them for a new one! A solid, reinforced slab upon which both abutments rest, as mentioned above, is not likely to become undermined. However, it is important to remember that soft, compressible soil is also easily washed away. Historically, stone bridges founded on wooden piles seem to have had no end of scour problems. Flaring the base of the bridge helps prevent settling, but offers only a little protection from scour. Scour is an important consideration to keep in mind when building a stone bridge on soft ground.