Choose a spot for the stone fountain that is close to both a water source and an electrical source. Avoid areas that have a large number or trees or other falling debris that may clog the fountain pump.
Lay a plastic pond basin on the ground and draw an outline with spray paint or some other material to show you where and how wide to dig the hole. Use a shovel to dig out the area, 2 inches deeper than the height of the basin. Also dig a trench approximately 2 inches deep to a nearby electrical outlet.
Pour crushed stone or pea gravel into the newly dug hole to a height of approximately 2 inches. Compact the stone with a tamping tool or scrap piece of lumber. Place the basin on top of the stone and into the hole.
Set 1-inch-diameter PVC pipe in the trench leading to the electrical outlet. Feed the electrical cord for the pump into the pipe and lay it in the trench. Cover the PVC pipe and trench with soil.
Place the fountain pump at the center of the basin. Cut a piece of metal mesh screen material with tin snips to fit over the basin and 1 inch beyond on all sides. Cut a small hole in the mesh screen to run the electrical cord through and a 1-inch hole directly above the pump for the water outlet.
Cut a length of 5/8-inch copper pipe with a pipe cutter long enough reach up from the pump to the top of the planned stacked stones. Attach a 5/8-inch compression fitting to the copper and tighten it with an adjustable wrench. Pass the pipe though the hole in the mesh screen to hand tighten it to the pump outlet. Lift the screen and secure the two together completely.
Fill the area around the pump at the bottom of the basin with pea gravel to give the fountain a more realistic look. Make sure that the gravel does not obstruct the water intake port on the pump.
Arrange the stones in the stacked order best suited to the look of the fountain you want. Remove the first stone and lay it on a concrete block or brick to keep it steady. Mark the location of the center of the stone with a permanent marker. Attach a 3/4-inch diameter masonry bit to a power drill and bore a hole through the rock. Apply equal downward pressure to get a straight hole; have an assistant pour water over the stone while drilling to protect the bit. Repeat this with all other stones, reversing the stacked order as you go.
Cut strips of 1-by-3 treated lumber with a circular saw, long enough to lie across the basin and screen and extend over the edge an additional 1 1/2 inches on both ends. Cut enough strips so they can be laid parallel all the way across with a 3-inch gap between each pair. Lay the wood over the screen.
Place the bottom-most stone over the copper pipe and down onto the wood strips. Continue stacking stones until they are all in position. Adjust the layout by turning the stones as desired and place smaller stones into gaps to give the fountain a natural look of weathered stones. Place a final two or three smaller stones at the top to obscure the copper pipe without blocking the opening.
Add water to the basin with a garden hose so the water is between 3 and 5 inches above the pump. Plug the end of the pump's power cord into an outlet and check the flow of water over the stones. Make any necessary adjustments and begin placing an assortment of other stones over the remainder of the wood strips and basin to hide it.